Here I’m constructing a list and analysis of existing RPG’s innovations, unusual twists, and mechanical “bells and whistles”, some of them fairly obscure, partly to clarify my own thoughts but perhaps of interest to other would-be designers.
A given idea may actually be reused across a number of RPGs, so readers may find weird systems mentioned with regard to what's still actually a familiar idea. As this is intended to be understood by the most readers for design purposes rather than being a historical reference, I usually list the most common game using an idea (if I'm familiar with it) rather than the first. Where warranted fairly obscure games are discussed.
Anyone who wishes to discuss anything feel free to do so (honestly I expect little input, since tumbleweeds hate game design). I’m inputting data topic-by-topic for clarity. If you haven't visited for awhile, there's a good chance I've edited in additional notes since then.

Looking back on it, I do need to apologize for this thread not being the best organized; roughly speaking the first few pages deal with character generation, then core mechanics, then combat, then miscellaneous things, then powers/magic. If looking for notes on a specific topic or system I recommend using 'Search this thread' (near top of page, if logged in). Thanks to everyone who participated, anyone who does read this in future, by all means if you see an omission feel free to reply to the thread or PM me.

8 June 2014 - Added summary of contents here for ease of use (Only 3 years late...PS when designing your RPG don't forget you need an index... )

Summary of Contents with hyperlinks

Page 1 - Character Generation
Attributes - generation
Attribute Effects
Over the Limit (attribute score range)(Page 7)
Races
Classes & Skills
More On Skills (#, ability modifiers, specializations)
Advantages/Disadvantages (see pg. 12 for more on point costs)
Improvement
Character Design Interactions with Core Mechanics

Page 2 - Dice Mechanics
D100s
Additive Rolling
Multidie additive
Roll Under
Changing-Dice-Type (formerly 'step dice')
Universal Table
Dice Pools


Page 3 - more on Dice Mechanics
Dice Pools (take-highest)
Exotic dice mechanics
Non-dice randomization
Multiple resolution systems and the trend toward universalization
Effect
Cutting Down Excess Rolling
Extended Actions
(see page 12 for opposed rolls, open-ended and impossible rolls, bonus & penalty accumulation)
Safety Valves (i.e. luck points)

Page 4 - Combat
initiative /round structure
Actions per round
Hit Points
Damage
Critical Hits (page 14)
Armour
Movement
Movement & tactical combat further discussion (Skarg) (Page 23/ linked thread)

Page 5 - more Combat
Core mechanics & combat manuevers
Combat Moves (multiple posts)
      -Part I Basic Attacks, Attack + Movement Actions, Multiple Actions
      -Part II - Movement, Defense, Cooperative Actions
      -Part III - Initiative Actions, Damage Modifying Moves, Special Attacks, Recovery Actions, Miscellaneous
Terrain & Environment


Page 6 - general adventuring
Vehicles
Adventuring Situations e.g. saves, encumberance, luck rolls, starvation/thirst
Perception
Craft & Repair    


Page 7 - general campaign & Misc.
Social Checks  
Equipment & Currency
Morality        
Weapon Proficiencies & Strength Requirements (and weapons design)
Super Powers  

Page 8 - Powers, Magic        
Powers List
Power Advantages & Limitations
Psionics
Chi
Magic - underlying concepts
Magic - who can use it

Page 9 - More Magic
Magic- spell failure
Spells per day
Magical skills,
Varieties of Magic

Page 10 - Still More Magic, Misc.
Miscellaneous Magic notes
Monsters
NPCs

Page 12 - back to resolution systems mostly
More HP/damage/injury systems (RobMuadib)
Open-ended & impossible rolls
Contested Actions
Constrained Design Spaces
Controlling Bonus & Penalty Accumulation
Point systems & costs

Page 13
Unknowns in design              
Sample Online Probability Calculators

Page 14
Conceptual Scope

Page 15
Complexity (moved from pg. 17)
Thought Processes In Designing / Hybridization How-To / Conversion between systems

Page 16
Divine Ascension

Page 17
Alternate Realities and the DRF (J Arcane)

Page 18
Types of derived attribute (?)

Page 19
Resource Management
Exploding dice probabilities (ggroy)

Page 20
Abstraction level
Edges & centres (link) (Clash)

Where a note refers to hypothetical speculative mechanisms or my own designing, rather than an existing game, the note is being rendered in italics.
An (*) under edit notes refers to a note not incorporated into the thread's document version.

Attributes:
 
An attribute (aka ability score, stat) represents a character's inherent ability at a range of related tasks. Attributes are normally possessed by all characters - at least all player characters - and often all monsters as well. A few games may have "customizable" attributes (i.e. FUDGE had sets flexible from campaign to campaign; Fuzion also suggests adding extra attributes for some campaigns if desired and then giving out more stat points to buy them). More rarely an attribute set may vary from character to character - e.g. where a roll has a default and various attributes are potentially definable for characters (Over The Edge, Cortex +) (thanks to RobMaudib for noting these). A system may let a character use a default rating if they have no applicable rating; this works particularly well with attributes rated positive/negative around an average of 0.
Certain attributes may also be used for some inanimate objects e.g. a bridge might have a passive STR for how much weight it can support, or a door for how hard it is to break down; most objects have some form of durability and vehicles will have some sort of Speed rating. Mechanical sources of damage (spear traps and the like) may need 'Str' scores for modifying damage, or this may be assumed to be built in to base damage.
See the posts on 'NPCs' and 'Monsters' for discussion of games where those lack scores.
 
Number of attributes used in a system varies dramatically i.e..
0- Wilderness of Mirrors, FATE, The Agency (see last section of post), ZeFRS (see 'untrained skill use' for notes).
'Searchers of the Unknown' for AD&D ditches all attributes to just use a monster stat block (AC, MV, HD, hp, #attacks, damage), though it does roll hit points...effectively a single stat of 1-8 [all HD reroll on level-up), Mutant Bikers (see last section).
1- TWERPS (Strength, apparently used for everything - stat checks, combat, HP).
'Over the Edge' lets players choose freeform traits, but arguably also has one 'attribute' (despite different characters skinning it differently) inasmuch as one trait must incorporate fighting ability, and is used to set hit points.
2 - Prince Valiant; 3:16 (in both cases the two are essentially 'Fighting Ability' and 'Non-Fighting Ability'). Supergame 3E has 'Fitness' and 'Insight' (physical and mental).
3 - TriStat (Mind, Body, Soul), The Fantasy Trip (GURPS predecessor)(STR, DEX, INT)(just two, STR and DEX in original MELEE book); Fighting Fantasy (Skill, Stamina, Luck); Lejendary Adventures (Health [including body and mind], Precision, Speed)
4- GURPS (Strength, Intelligence, Health, Dexterity); Amber (Strength, Warfare, Endurance, Psyche)
5- Savage Worlds (Strength, Vigour, Agility, Smarts, Spirit)
6- D&D (Str, Dex, Con, Int, Wis, Cha)
Games with 6 attributes will sometimes determine randomly a stat affected (e.g. by a curse) with 1d6.
7- TORG (Str, Dex, Toughness, Mind, Spirit, Charisma, Perception)
8- Palladium (Strength, Prowess, Phy. Endurance, Intelligence, Mental Endurance, Men. Affinity, Beauty, Spd)
9- Storyteller (Str, Dex, Sta, Int, Wits, Per, Cha, Manipulation, App. ), DC Heroes (Str, Dex, Body, Int, Mind, Will, Influence, Aura, Spirit)
10- Fuzion (Int, Will, Presence, Technique, Reflexes, Dexterity, Con, Str, Body, Movement); Rolemaster
11- Harnmaster (Str, Sta, Dex, Agility, Eyesight, Hearing, Smell, Voice, Int, Aura, Will  (13 if you want to count Frame and Morality).
14 - Space Opera (Physique, Str, Con, Agi, Dex, Int, Intuition, Leadership, Bravery, Empathy, Psionics, Technical Aptitude, Mech. Aptitude, Electronic Aptitude
 
Balance between attributes can be a delicate affair where number of stats are low, though these aren't (IMHO) inherently broken. While having 4 stats that are all used for 1/4 of skills/game functions should work...there are are however examples of systems with low numbers of stats that are quite min/maxable - GURPS for example being originally designed for fantasy, STR is important there for hand-to-hand damage etc, but becomes much less valuable in settings as soon as firearms (or blasters) make an appearance, while DEX remains invaluable. GURPS 4E reduces cost of STR and Health compared to Int/Dex, and uses STR to determine hit points (though perhaps this makes Health the new dump stat..)..Tri-stat reputedly is fairly easily broken as well. Random roll systems are less susceptible to breakage, though more stats makes a series of lucky rolls less likely. Many games end up having a "god stat" which dominates, while class-based games (D&D) often have class functions based off a 'prime requisite' that make it critical to members of that class. Passively operating stats (Constitution) are sometimes disliked as being dull.
 
Higher numbers of attributes mean each individual stat is less likely to give overwhelming advantage. However, more stats does usually mean more points, so that it becomes easy to max out one or two attributes with relatively little opportunity cost (by spreading out the effects of dumping across several stats).
 
Games with low numbers of stats tend to have stats which are 'fuzzy' in terms of interpretation, while more stats means each becomes more specific in what it measures. D&Ds 6 attributes for instance are all quite vague (Dexterity combines hand/eye coordination and agility, Strength includes size and muscle, Wisdom includes willpower and judgment, Charisma can include looks and personality), which has the advantage that a player can choose to interpret a number in a variety of ways to support different character concepts, while Size or Luck or Willpower are harder to interpret creatively, but are clear and measurable.
A lot of very specific attributes can lead to checks having to be modified by more than one attribute (e.g. HarnMaster).
 
 
A fairly comprehensive listing of attribute sets for a number of RPGs is available at:
https://web.archive.org/web/20130730183539/http://www.fudgery.net/omnium-gatherum/lag.html
 
Some of my favourite obscure attributes are Size (most commonly seen in the Runequest/BRP family of systems, very useful for modelling a number of things like lifting objects and hit points), Speed (frequently handled messily) and Power (as seen in RQ, SenZar and Stormbringer, this stat is handy if games are going to have a lot of soul damage/soul eating going on, given a working system for attribute damage). In games not treating Size as an attribute, it often modifies attributes such as Str; in other systems height/weight may be figured backwards from the Strength score, or Str may be explained to be a relative value (to other creatures of the same size) with Size and Strength separately modifying physical tasks. SpaceMaster has a Psionics attribute which is a potentially interesting idea. Tunnels and Trolls has a 'Luck' attribute; one interesting idea from MSPE based on that is designing background to factor in how (un)lucky a character is, giving low Luck characters more bad events (another idea I'd toyed with was for a games with random background and rerolls based on Luck, having a character be able to spend those in random background generation). Another common-ish idea is splitting up "Dexterity" into an actual Dexterity (hand/eye coordination) and Agility (as these may be very different).
 
Other innovations, good or bad include:
-subabilities: some systems divide each stat into multiple substats which are related. 2E D&D in the “skills and powers” era divided each stat into 2 subabilities (an idea originally from Star Frontiers); one could be raised by lowering the other, or one would start equal to the other, with a random variation of +/-d4. This tended to be broken since often one subability mainly affected one classes' abilities (such as spellcasting) and could be dumped by other classes to raise the other which was often generally useful, and since the AD&D stat tables were built around non-linear progression - S&P fighters with Str 17 could have 'Muscle' 19, going from +1 on damage to +7.
Nexus the infinite city had three attributes (cost 10 pts/level) with 4 subattributes each (3-5 pts/level); it also lets characters 'pair' subattributes (e.g. Bravado = Charisma & Will, Physique = Strength & Toughness, Quickness is Speed & Move) which gives a cost discount for raising both at once (5 for both).  Feng Shui kept this setup for stats although the stats are set by archetype and might as well be 12 entirely separate stats. GURPS also has this via advantages (e.g. “split” STR for taking damage vs. everything else). TriStat dX has a "Less Capable" defect for creating e.g. characters that are undextrous despite high BODY; this is fairly handy since it has only three stats.
Subabilities can also exist in a sort of ad hoc fashion e.g. AD&D also had separate Con values for resurrection (original score only) and everything else, or Rifts Atlantis has a giant squid with outrageous lifting capacity (like 60,000 tons) but relatively low tentacle damage.
SenZar has a number of special powers many of which have 'mods' (+1 to +5) adding directly to a stat for some specific purpose, such as 'Eagle Eyed' and Perception, or Seduction to Presence; it has the issue that mods often cost the same as full attribute points, for a stat of 11 or less.
Skills can also provide a similar function in some games.
Subabilities main purpose is to allow characters or monsters that are good at [subability] even though their general attribute is low. If the subabilities line up mostly with skills this can be accomplished with less complexity by having attribute give a higher base skill (e.g. D6 Star Wars: a low score can still be bought up to a high value) or a skill purchase discount (Savage Worlds) rather than a separate modifier (a la d20 system), although the former two options do make cross-matching attributes and skills much more awkward. See also post #8 ("attributes and skills"). Whether a 'use skill rating in place of attribute' works to replace a subability however would depend somewhat on how the various system numbers are scaled. Arguably it may generally be better to just have more attributes to start with rather than fiddling around with 'subabilities'.
 
- groupings of attributes: a number of systems including DC Heroes, Fuzion and the the Storyteller system, have broad categories within the attributes (such as “social attributes” or “active attributes"). This may help in determining which attribute should apply to a given task, but also IMHO tends to give large sets of attributes and/or strange edge cases or extra attributes (like DC Heroes using INT/WILL for Perception checks; Storyteller having Willpower as a rating distinct from the normal attributes). Storyteller uses its grouping largely to generate attributes (split 7/5/3 between the categories initially) while DC Heroes uses the groupings to determine which attributes apply to tasks on its action table, with all the categories needed to generate consistently-scaled resistance and effect values. Amazing Engine paired stats in character generation; multiple characters generated from the same "player core" had the same base number of dice to distribute but could change which stat of the pair they prioritized, the idea being that players would play similar-but-not-identical characters when/if the game moved to another AE setting. Task-resolution where a roll is directly against [attack score - resistance score] can also generate a need for some extra stats to exist so they can follow a different range e.g. Dallas RPG (see contested actions).
-attributes could also be split into 'primary' (important) and 'secondary' (less important) groups e.g. in Leading Edge Games' Aliens game there are several 'secondary' attributes which are always rolled (3d6), though primaries can be point-buy (or are 3d6 twice take highest); other systems use this less overtly e.g. DragonQuest has point buy for primary stats but rolls Appearance (same scale but no point exchange is allowed due to App's lower value), as does 1E D&D with Unearthed Arcana comeliness. MERP has an Appearance (d100+Presence primary stat bonus), which can be switched with a primary stat with GM approval, but usually only if APP is raised & primary is lowered.
 
Slightly related to 'grouping' of attributes, an rpgnet thread awhile back discussed the ordering of the ability scores in D&D. OD&D wrote scores in order Strength, Intelligence, Wisdom, Constitution, Dexterity, Charisma (the 3 prime requisites for main classes + secondary attributes); 1E added thief and so went Strength, Intelligence, Wisdom, Dexterity, Constitution, Charisma (4 primes + secondary), 3rd ed went Str/Dex/Con/Int/Wis/Cha (physicals then mentals), 4th edition had order Str/Con/Dex/Int/Wis/Cha (similar, but also ordered in pairs for determining defenses i.e. Str and Con both determine Fortitude defense).

-attribute customization - an attribute's definition and hence what its modifiers affect may vary. In simpler RPGs a high/low stat may generate fiction which adjusts if a modifier will be used, circumstantially (i.e. in 0D&D if a character with a low CHA is ugly they might ignore some of the penalty by wearing a bag over their head, which wouldn't have helped if their lack of Charisma was actually due to an annoying personality; David "Zeb" Cook in the 2E AD&D PHB suggests a Dex 3 character might be 'naturally clumsy or as blind as a bat', but here is (he remarks elsewhere) reminiscing about an 0D&D character he knew - the idea doesn't quite work in 2E without much DM handwaving, mostly ad hoc penalties - the Dex adjusts range penalties and surprise, but 'perception' checks in 2E more often default to Wis and/or Int. Systems with more complex rules may allow for formal adjustment to what a stat modifies i.e. in 3E D&D a character with low DEX might still be a crack shot due to Zen Archery, letting them apply their Wisdom modifier to ranged attacks instead. The oWoD Storyteller system allowed characters with very high stats (4+ on a 1-5 scale) to choose 'specialties' which let them reroll 10s on specific related tasks i.e. a character with Appearance 4 might have 'busty' or 'exotic', while a character with Intelligence 4 might have 'visual memory' or 'analytical'. Hero System lets a character apply some power modifiers to stats e.g. to increase knockback for Str.
Another interesting example is in Dungeon Crawl Classics, where every character has a LUCK score but where what rolls this modifies is largely determined by a 'birth augur' table - a character who 'survived plague' adds their Luck modifier to magical healing, while a character who 'survived a spider bite' adds it to saves vs. poison (the stat also has some universal uses e.g. vs. critical hits and for spending in play). The augur roll description is slightly at odds with low score effects (e.g. a character with a low Luck who is 'born on a battlefield' would have a damage penalty, rather than bonus). Its perhaps odd realistically, if balanced, that a character can't have 0 or 2+ augurs (if you're born on the battlefield you can't also survive plague). Probably a roll should only be required if the character has positive Luck.
-attribute effects may be 'customized' by race, like AD&D dwarves having a poison/magic save bonus of +1 per 3 1/2 Con other characters don't get.
-another idea here customizes attributes by 'class'; every character might have a Body attribute with the Str-fighter adding this to more physical rolls and the swashbuckler to dodging.
-(another form of 'customization' that might be useful would be to have a generic 'power' attribute that reskins to different classes in different ways - magical power for a mage, psionic power for a psion, etc. - with effects like determining power points or modifying class-related skills. Having this separate would let this be balanced better than the usual setup where different classes get their bonuses from a normal attribute instead, blowing out its importance and e.g. making all wizard characters have 18 Int). Depending on how you look at it, you could think of the core archetype power ratings for the oWoD storyteller games [Generation, Arete, Rank] as being sort of like this across their various games, although they're not considered 'attributes' exactly by their home systems)

You might expect games that are attribute-based and using class to use more either extra attributes for certain classes or attribute customization to differentiate classes, but I can't think of many examples here, perhaps because its an unusual pattern. An early 'Sorcerors Apprentice' magazine added a 'Piety' spellcasting stat for use with an optional Priest class for Tunnels and Trolls, while WHFR has non-zero magic ratings for wizards only. Class features or skills that require a raw attribute check to function might be said to give some 'redefinition' to an attribute.
Primarily attribute-based games with classes can encourage certain classes to get bonuses to their key stats. Warhammer 1E/2E, while not exactly a class system, has 'advancement schemes' that give warrior types more 'weapon skill' attribute increases in their profiles.

-extra attributes for certain characters. For instance, original Fighting Fantasy had the option of playing magic-users in an expansion book; these rolled up a 'Magic' score giving spells per day in addition to the normal Skill, Stamina and Luck (which suffered penalties). 1E Shadowrun had 'Magic' scores for magic-users, although this was nearly always (barring rare effects) the same as Essence which everyone had (default 6 rather than rolling). The Golden version of d20 (see notes under advantages) had feats allowing characters to roll up extra stats.

Games can also have powers or skills that operate on the same scale as attributes, letting these function as extra 'attributes' if needed despite not being defined as such (Marvel Super Heroes, DC Heroes, Savage Worlds).
(From a purely technical standpoint you could say that an 'attribute' is really something all characters have, and in that sense maybe a supernumery attribute isn't really an attribute. However, within the context of their game systems these function just like the ubiquitous stats, so I find the term reasonably useful still).

Thanks RobMaudib for some additional discussion

Unusual Attribute Arrays
Systems often have a few oddities in their attribute structure, as a consequence of their evolution or of attribute 'grouping'.
-Icons has similarly named but very different stats  "Determination" and "Willpower" - Determination giving points for roll bonuses while Willpower is for resisting mental attacks. It inherits one from parent system MSH; the other developed for dealing with FATE aspects.
-Storyteller has a Willpower (for point spending) separate from its 9 basic attributes that are forcibly fitted into a 3x3 conceptual grid leaving no 'room' for it. (it is initially calculated as a derived attribute in some forms e.g. nWoD Resolve + Composure, but can subsequently increase; in other versions e.g. original Mage it has a set base, varying largely due to 'freebie point' spend).
-D&D uses Wisdom for perception checks in 3E, something not consistently handled by any stat previously. Keeping the traditional 6 scores shoehorns this in along with its use for determining willpower and judgment.
-Cadillacs and Dinosaurs (T2000) Agility does not include manual dexterity; Archery and Firearms are STR-based skills.
-Maelstrom uses raw attribute checks for tasks, and has 'attack skill' and 'defense skill' as two of its nine attributes (there are also separate Speed, Agility + Knowledge, Will, Persuasion, Endurance, Perception. Point budgets mean that a higher Attack character will, in general, have slightly lower Speed/Agility. Actually, HERO is somewhat the same in that offensive and defensive combat values (OCV and DCV) are separate, although those are IIRC not defined as 'attributes' exactly. Interlock and Fuzion have separate combat and non-combat dexterity (Technical or Technique vs. Reflexes); Marvel FASERIP has separate 'fighting' and 'agility'. In most cases this is to prevent Dex being overpowered (a 'god stat').

Minimizing Numbers of Attributes
This can be handled by bundling several functions together into one stat, handling divergence between characters as a Skill (e.g. FATE Physique) or Advantage (GURPS "Strong Will"). Some injury systems (specific injuries/wound boxes) reduce need for a 'CON' or similar type stat. Stats can also be replaced with fixed numbers e.g. Movement Rate or Size based on species, or a fixed # of 'bennies' instead of having a variable Luck stat.
You could imagine a system which made high attributes a skill special ability, for instance a damage bonus from a high 'Fighting' type skill might be described as due to beefiness (and add to might-related physical rolls too) or as due to finesse and instead help on more Dex-like checks.
Another interesting approach doing away with attributes is for free rpg 'The Agency' (supernatural 60s spy game). Characters here just have skills; most skills default to a level of 'normal' (2 dice), but a character gets 1-2 skills that are 'poor' (=1 dice), representing a weak area e.g. athletics for an intellectual character, survival for an aristocrat, or science for the man-of-action type.
Mutant Bikers of the Atomic Wastelands (FUDGE) likewise has no stats. Some skills (Take It Like A Man, Interact with Others, Notice Things Happening, Move Real Fast, Figure Things Out) describe the same areas as attributes, but with no indirect modifications e.g. Figure Things Out won't improve 'Fix Broken Stuff' as well. Some Gifts may also be attribute-like.

Preference for stat-minimization climaxes in the Death To Ability Scores! movement (DTAS). A argument by some gamers (particularly 4E D&D players) is that ability scores (attributes) are a bad idea for various reasons. For instance (from here):

QuoteDo you really like using ability scores? Or are you just used to them? Because it seems as though you only want ability scores to be worth a +1 or +2 bonus at most and reduce them to vestigial bits. Put bluntly, it's not worth keeping the mechanical and meta-fictional baggage of ability scores if they're going to provide only a token bonus.
It's not worth the system mastery trap of needing to pick the sneaky class, the sneaky score, and the sneaky skill to have a sneaky guy worth a damn. It's not worth the twin bugbears of being mechanically punished for having a socially skilled fighter or true Vancian wizard (ie, a well-rounded adventurer like wizards actually are in Jack Vance's Dying Earth books); or being called a munchkin power-gamer because your fighter has to have an 18 STR to not suck but you roleplay your 8 CHA dude as a generally pleasant person. It's not worth watching the light go out of a new player's eyes because they wanted to play a guy based on their favorite fictional character but the realities of ability scores make it impossible.

All of the specific concerns here are fixeable by various means - if they are a problem - but this may be useful to keep in mind. In related design, storygame 'A Wilderness of Mirrors' avoids ability scores partly for that reason (saying super-spies should have all 18s).
See also: post #67 Over the Limit deals with difficulties of representing large ability scores

recent edits: Mutant Bikers note (*).; TORG gave stat list

Generating an attribute score usually falls into two categories: Random determination (die roll) or Point Buy. Some attributes may instead have a fixed base (e.g. Essence in 1E Shadowrun =6 base minus cyberware, or movement rate may be treated as an attribute but derived from race). Archetype systems (Feng Shui, TORG, Shadowrun) also sidestep either with prebuilt characters.
 
Point-buy is sometimes only for attributes, while in other systems the same pool may be used to purchase powers, skills, advantages/disadvantages, etc. Random roll is faster, but less “fair” and potentially abusable by cheating etc. The more attributes a game has, the less likely statistically that random-roll will generate a super character and the more involved a point buy system becomes. Where a system is fully random, there is less pressure for attributes to be equal in game importance and houseruling in additional attributes is easier (often seen e.g. with Tunnels and Trolls). (Palladium is also an interesting case: being 3d6 in order Strength being weaker in modern games where guns are suddenly available, is less of a problem).
A random system also allows rolling of minor non-attribute-related stuff, for instance, a random chance of a character having 'noble birth', being ambidextrous, or being psionic, is more acceptable in a random system than if everything was balanced apart from that roll, since a characters total value is much harder to quantify. Random systems also give an idea to GMs of how common high or low scores should be for NPCs which may not be as clear for point-buy games (particularly if the GM is allowed to build NPCs with any number of points). Random games usually limit the amount of bonus from attributes somewhat (e.g. 0D&D). There are sometimes rules for re-rolling 'hopeless' characters, or this can be left to GM fiat, or discouraged.
In games where a lot of player skill is involved (old school games), surviving with low attributes can be seen as 'hard mode' and something of a badge of honor (e.g. cf. Man Rider story here).  

Point buy systems quite often add slightly less logical attribute effects to attributes, to balance their relative importance, often particularly noticeable in systems which have evolved from random to point-buy across multiple editions. Some point buy systems (e.g. DC Heroes, HERO) charge different numbers of purchase points/stat point, depending on the attribute, thus avoiding both balance and realism issues but adding further accounting to chargen. While a good point buy system makes character generation theoretically fairer, a bad point-buy system may not be terribly fair, with some character concepts at a disadvantage if stats are same cost but different importance (e.g. to a given class) or if breakpoints are easily exploited. Even bad point-buy does let players get the characters they want though, within budget, which purely random rolling doesn't.
 
Given that both point-buy and random-roll have pros and cons, it is naturally harder to create a system that allows both as options, as this sacrifices some of the advantages of both systems. A 'mixed' system will need to be designed to meet point-buy needs not necessary for random-roll (such as all attributes being equal value), and would have to compromise on # attributes, and the need to have wide bands of similar ability vs. eliminating abuseable breakpoints. Random rolls for things outside the attribute system might need to be converted to point-buy as well, and some sorts of roll modifiers (like races giving re-rolls, e.g. a la T&T Anniversary Edition below) may be harder to duplicate or balance in point buy.

A mix of random and point-bought characters in the same party is almost as unequal balance-wise as if everyone were generating stats randomly (the odds of a huge discrepancy falls as number of randomizing players falls), although it still gives a player more opportunity to create the character they want. Point-buy and random-roll are usually geared to give out similar point totals, perhaps favouring random slightly unless a player is expected to roll multiple times. (one, 2nd Ed. Advanced Fighting Fantasy, assumes point buy with random averaging much lower totals)
 
More on Point Buy
Point buy may be at a linear cost, 1:1 up to a given threshold and 2:1 or more past that (e.g. Dying Earth, and 6E Hero lets the GM set a soft cap of their choice as a campaign houserule), require points equal to current rating, or require use of a table to calculate the scores. A fairly common design flaw of many point-based systems is to shift between using a linear cost in chargen, to a non-linear cost in character advancement, or vice versa, creating trap options. For instance Storyteller goes from linear cost to ‘triangular’ [current stat x 5], while 3.5E D&D goes from requiring [points equal to current modifier, minimum 1] to raise to a fixed +1 per 4 levels. A fighter with Str 17, Charisma 8 who advances to 4th level can add +1 to the Strength (worth 3 points) or +1 to their Cha (worth 1 point). In the opposite case, a storyteller character who wants to raise their attributes with experience should leave some of their stats at one dot to make the process cheaper; most pronounced in oWoD Vampire, whereas Mage  and I expect later games often uses [new stat]*x rather than [old stat]*x.
A possibly interesting rpg.net thread on this here: http://forum.rpg.net/showthread.php?713092-Rant-Linear-chargen-points-quadratic-experience-points-Whyyyy/page1  discusses why its off used (complex costs are too annoying when building every number on a character at once) and has a proposed solution (pregenerated arrays), implemented for nWoD here. The system also moves to linear costing in 'The God Machine Chronicles'.
This same problem can of course apply to buying skills or other abilities as well. Of the White Wolf games Aberrant is perhaps the worst for multiple resource pools with 'nova points', 'freebie points', 'stat points' and then XP all having different costs. Prebuilt 'archetypes' make it easier to fit out starting characters despite complex costing, but actually make linear-then-nonlinear costing worse by hard-coding trap options as standard careers and so reducing player choice.
FantasyCraft makes deliberate use of non-linear costing by having abilities which are '+1 to whichever is lower of two listed stats, choice if equal' - meaning that a higher value boost can be gotten if a character spends more points in a secondary attribute as well.
Linear-cost stat purchasing is simpler but doesn't help players' tendency for players to max out the main attributes for their class/race much. It is most viable if there are few points to distribute (Savage Worlds), if a grid is used (Storyteller), or if 'dimishing returns' is fundamentally built into the core resolution mechanic i.e. HERO characteristic rolls, and GURPS 4E use 3d6-roll-under meaning that the same number of points generates less and less return in terms of increase in chance of success. Linear systems often also have an un-even scale, where an 'average' score is less than half the maximum, meaning that lowering one score to minimum isn't enough to raise another to max. Both Storyteller (1-5, but with 2 rather than 3 being considered 'average') and Savage Worlds (the same 1-5 scale but described as dice types so d6 average, with 1 step below for d4 and 3 steps above for d8/d10/d12) do this.
Non-linear cost point buy gets increasingly annoying for larger scales of attribute numbers.

Point buy elaborations:
*Amber Diceless uses an Auction in which characters compete to purchase values in the games’ attributes. The number of points spent on an attribute in the auction is permanently recorded, but the number itself functions as an ordinal rather than cardinal/fixed quantity; it has few if any rules-specified modifiers or effects, only being used for comparison against other players (highest score wins). Players who do not bid may reduce their base scores below 'amber rank' to 'Chaos Rank' or 'Human Rank', which do have some defined debilitative effects. The auction may complicate bringing in new players/replacing dead PCs.
*vaguely similarly, "Cop Show" has a chargen system where players go around in a circle, and what each player specifies they're good at makes the next player bad at it. http://www.criticalmiss.com/issue11/BlockBuster.html

*3.x D&D suggests just giving out a standard "array" of scores (thanks Beejazz) i.e. 15,14,13,12,10, 8 for PCs (the "elite" array) as something simpler than points. Its interesting to note that if you're doing this for 5E, the number of 'odds' in the array should be chosen carefully as characters have different numbers of +1s to raise stats (a normal human gets six +1s, a feat human two, and most nonhumans one - therefore you need three odd scores before a standard human is worthwhile compared to a "feat human". Where the odds are might also give an interesting choice between boosting highest or reinforcing a weakness (e.g. I quite like 18,17,16,15,14,13 for a high-power game).

*A few point systems give players bonus points for stuff such as detailing the character's back history or volunteering to write a campaign journal (DC Heroes, Risus, Amber).

*Shadowrun 1E is effectively point-based but points in various categories are determined from a Priority Grid.
The (slightly descended from it) Storyteller game uses basic attribute groupings (Mental/Physical/Social) to generate scores - starting at 1 dot in each and adding additional 7, 5 or 3 dots depending on priority (max. 5), in order to constrain min/maxing.
*3.x D&D uses point-buy with a "standard array" of pre-bought scores often used to speed up character building e.g. PCs/elite NPCs may use [15,14,13,12,10,8], equivalent to 25 point buy. Apocalypse World gives each class a choice of several arrays.

*some point-buy systems cost different attributes differently based on their importance (HERO).

*a couple of games let one attribute start with a higher maximum than the others. Mutant:Year Zero does this based off role (class?) 'key attribute' - this can go to 5, while other stats go to max. 4, when splitting points; Pathfinder and 4E D&D do this indirectly, give humans a racial bonus to a stat of their choice after points are allocated, so that one stat goes to 20 rather than 18. Apocalypse World has a stat advance related to class as one possible option initially. All these keeps maxed attributes slightly rarer (by limiting them to a specific class, and one per character, so perhaps characters are more diverse).

*The old Conan RPG from TSR (cloned as ZeFRS) has a system where characters don't truly have attributes, but where a character buys skills and their ranks/10 determine a 'default' for other skills in that group (an interesting idea, but with its table-based system the tiny increases in default were hardly worth tracking).

*Fuzion rates stats 1-10, with a 2 being an 'everyday' person score. Cost is linear. Normal humans are limited to a 7-8 but 'superhumans' can have more than 10, and with GM approval a character can put whatever number of points into scores (as written this could be 50+ though probably not intentionally).

*5E D&D lets character either random-roll or point-buy characters, but point buy can't purchase a base score over 15 on a 3-18 scale.

*Two-Fisted Tales uses a 1:1 linear cost for attributes, but one attribute "Weird" determines attribute max. (for other scores) and so needs to be increased to buy up other stats to especially high levels.

*Freerpg hi/lo heroes has a set of stats which can be either 'high' or 'low'. A player chooses Build (Quick or Powerful), Mentality (Logical or Intuitive) and Temperament (Bold or Cautious) which sets all their stat values e.g. a Powerful character will be high for Damage but low for Move.

*The Pathfinder 2E playtest instead of giving out points gives raises (each worth +2 if a score is up to 18, or +1 if a score is higher). Each character gets a number of increases that can be applied to any stat, plus raises limiting to one or two particular stats for race, class, and 'background'. If random rolling instead this replaces most of the raises (not some racial increases). Overall effect is somewhat similar to FantasyCraft or perhaps 13th Age but sort of more 'granular' with the raises rather than exact points.

More On Random Rolls
Random rolls are generally a bell curve roll such as 3d6 (by tradition and to duplicate IRL bell curves i.e. keep extraordinary scores rare). World of Synnibarr, Chivalry & Sorcery and "The Arduin Adventure" use d20 rolls for stat generation; "The Compleat Arduin" uses smaller linear rolls, such as d10+6, with exact dice/bonus depending on stat and race). Fantasy Imperium uses d100 (rolled under directly for checks); Rolemaster uses d100, but with the derived bonus modifiers following a bell curve. Toon uses a d6 roll for attributes (but characters can't die, probably have as much fun failing, and sometimes need to fail checks to get results).
It is theoretically possible to have a 1-100 system that follows a bell curve; dark legends exist of an RPG which uses the "Danish Lottery Dice" (d34) to generate a bell curve from 1-100 [3d34-2]. See http://forum.rpg.net/archive/index.php/t-60023.html (the thread there suggests the game in question might be Metascape, but this is definitely incorrect). (It is also possible to do a 1-100 steeper bell curve with 4d20+d24-4).
The boardgame (more or less) Flash Gordon & The Warriors of Mongo - see review here (thanks to Philip for mentioning this game elsewhere), apparently used 3 "averaging dice" to generate stats; these were common in wargames and are six-sided dice labelled (2,3,3,4,4,5).
A very few systems with random stat generation have 'rolling up' occurring as part of the random roll. These include Tunnels and Trolls (triples add and roll over in later editions; derivative Mercenaries, Spies and Private Eyes adds +2d6 on triples, once only), Palladium, or in a sense AD&D with its 18/percentile Strength. Palladium rolls up for a 16-18 on 3d6 or in the original Palladium Fantasy game 17-18, with 12 on 2d6 also granting +d6. Aliens Unlimited lets aliens with 4d6 roll up if they get a 16-18, but not if their roll is already higher. 12 on 2d6 is OK (letting disadvantaged races sometimes get the equivalent of a normal stat roll) but this is otherwise questionable...and would be equivalent mostly to just having a bonus table that scales up faster. Arguably since races rolling 4d6 or 5d6 at least don't roll up, it lets the system, rarely, create humans say with Troll-equivalent strength, I suppose. Mutant Epoch and TSR's Marvel Super Heroes have roll-on-a-table-determined stats which likewise vary tremendously, but with the attribute check system (table lookups) mitigating most game effects of high numbers (aside from raw damage in the case of Str in MSH).

Random generation is only rarely used by systems with small attribute ranges (like Storyteller's 1-5 scale), since generating the scores would possibly involve less dice (and so be very variable) and also since shifts in the number are more significant. Exceptions do exist like the GDW House System (1-10 range, rolled on 2d6-2, reroll if 0) or Anima; Beyond Fantasy (1-10 range, rolled on 1d10). A low-scale system could still have randomly-allocated points without being too imbalanced (i.e. with six attributes, roll d6 for an attribute point to see where it goes; roll again for each stat point in the spend pool - one heartbreaker 'Pits and Perils' does this, with a roll on a table to see which stat a character is 'Mighty' in). Another, 'dungeonrobber', has a separate d6 roll for each stat, with a 6 giving a 'High' rating. In general however, it seems that low scales such as 1-5 tend to have PCs often-to-always have some stats around the maximum level, making these sorts of scales more suited to the assumption the PCs are heroic/exceptional.
Point allocation is also less math-intensive with smaller attribute scores - in part games with low stat scales (i.e. 1-5) sometimes seem to have more attributes, perhaps partly due to this and partly due to the need to balance attribute effects by sub-dividing powerful stats.
Random rolling can be used in games with very few stats but its not a good idea. The worst system I've ever seen for this was the 2-page freerpg Dementia - roll 2d8 points and divide them across 8 attributes, and checks are under stat on d8.
A similar idea of point-buy with a random number of points can be seen in (I think badly misnamed) Dragon magazine #132 article "A Little Less Super" for DC Heroes 1E, which has a random 'origin' roll that adds from 0 to 10,000+ character generation points to the base of 250 ('no extraordinary origin' 0; mutation (minor) 250, scientific origin (major) 1,100; other-world origin 10,000; multi-origin roll d10 times). It rationalizes powerful characters as multi-origin e.g. explains that Superman has an 'other world' origin (Krypton), a 'chosen one' origin (destiny) and a scientific origin 'due to Krypton's advanced civilization'; while Cyborg has 'personal injury' and 'scientific'.

 Random variation in a single variable only also occurs in Over The Edge (hit points only) and in some 3.x D&D games with point-buy where the only random roll left is also for hit points; optional rules may however make that nonrandom as well. Ringworld (BRP based) has the usual RQ attributes but with a devastatingly important random roll for age that gives a huge variation in number of discretionary skill points.

Other random-roll elaborations:
*a few random-roll systems generate PC stats by making a random roll for each stat, then looking up result on a table e.g. Mutant Epoch, Star Frontiers, Marvel Super Heroes (all of these these use d%). This can be used to reduce odds of unworkably low scores, or likelihood of higher scores, or otherwise adjust the range as desired.
Games which normally use dice sometimes use a table in special circumstances: D&D 3E in the DMG (and 2E in 'High Level Campaigns' monsters with ability score rules) have tables for monster scores, so say a monster race with a 4 Int normally might vary from Int 2 to Int 6 based on the 3d6 roll and a table lookup.
*Unearthed Arcana (1E AD&D) has a class-specific random-roll system for humans giving different # of dice by stat e.g. a barbarian would get 9d6 for Str (take 3 highest) but only 3 for Charisma. (This also gives humans good stats). An oddity would be that since class is thus set when dice are rolled, it could create characters who would be better off dual-classing.
*2E AD&D (skills and powers IIRC) had a system whereby 24 dice are split between stats, with a minimum of 3d6 and maximum of 9d6. This gives results somewhat similar to UA's (instead of choosing from the arrays, you engineer your own) however with the differences being that class choice isn't set finally if the rolls don't go according to plan. However, more dice do add steadily diminishing increase in the average such that [all 4d6] is the optimum for total point production, 4d6-arrange-to-taste being probably slightly better.
*T&T 30th anniversary (Alternative Rules) the player rolls 18 d6s, then allocates 3 per ability. Some races may "take two worst rolls" for an ability, or reroll e.g. 5s and 6s (similar ideas could be applied to the Unearthed Arcana/2E rolling system, above). (Racial modifiers like this are more difficult to calculate the effect of though, and its particularly tricky to cross-apply the random effects onto a 'point buy' system and have characters from either system be equivalent).
 
*Grodzichi's "Low Fantasy Gaming" RPG gives out one 15 automatically, then rolls 4d6-drop-lowest for the other 6 stats. It also allows players to copy other players' rolls with a slight (1-3, GM discretion) point penalty.

*'Beyond the Wall' is random but using a 'lifepath' system to generate stats (along with some skills, etc), instead of rolling directly. Events can create links to (and bonuses for) other players' characters.

*Tunnels & Trolls normally does stats as 3d6 in order (sometimes with triples add and roll over), but house rules I've seen on the internet have also included tests of player skill such as shooting basketball hoops (get it in and get an 18; otherwise you get a 3), or beer-chugging. T&T gives high-level characters a sort of access to 'point buy' with the 19th level Omniflex spell, which lets the caster rearrange a target's attribute points between stats, as long as the total is unchanged (in 7th Ed. T&T, Omniflex can also be used to generate an infinite stat-gain loop together with the spell Unlucky Bees, which deals damage equal to the target's Luck score and then adds a point to permanent Luck if survived- shift the Luck points gained to other stats with Omniflex and repeat).

'The Gang Hack' (scooby doo) has a system of 3d6, where a roll of 15 means the next score is rolled on 2d6+2 to balance it.

*systems sometimes give bonuses to characters from previous characters e.g. Synnibarr and Dragonlance (both random roll) add bonuses for "dying heroically". Bushido (point buy?) was notable for adding part of a dead characters 'Honour' score to the new characters stat (encouraging heroic death). TSR's Amazing Engine let a character earning XP add it to either itself or to the "player core" it was generated from, for use in other settings. (Related to this AD&D Dark Sun had 'character trees' where a character advancing also improved other backup characters in the tree, but this actually didn't affect attribute scores directly - is more a rough analogue of 3E's bringing in replacement characters at the same level).
 
*Games can have rules for randomly generating characters based off parent's scores rather than wholly at random.
(this could be based off one set of known scores for a known PC score, or assume scores for both parents are known).
A 1-10 or dice pool game could do this pretty easily (for each stat, roll both parents' stats as a dice pool at a TN generating half as many successes on average; add results for both parents together), or a game might have a table including [as parent A/as parent B/slight modifier/roll randomly] type results [Talislanta essence extractors].

*2nd edition D&D generates attributes randomly, but with a variety of methods. Not all are balanced but assuming multiple methods are available, there are trade-offs between them. Method IV (4d6 drop lowest twelve times, take the six highest and arrange to taste) generates characters who are generally superior but rarely have 18s, while VI (base 8 for each stat, add 7d6 where desired) generates a couple of poor attributes and some very high scores) but not as severely as IX (divide 75 points as the player sees fit). IV characters tend to be better at raw attribute rolls (e.g. proficiency checks) with their higher scores whereas VI characters have better modifiers (hit, damage, HP, etc.), by having a couple of really high scores, high enough to get a bonus on its odd nonlinear stat charts. See here for discussion.
Early 2E with no optional rules is geared more toward the lower methods (I,II,III), while more optional rules and later books include some inflation, e.g. if you're using the Complete Fighter, it recommends using method VI (quasi-point buy with base 8 then add on 7 d6 rolls), since 'kits' add a further set of prerequisites on top of the default class and race minimums.
2E also includes prerolled 'arrays' by class in a number of optional class books (Complete Ranger, Paladin, etc.). Arrays are not necessarily balanced so some rolls (usually on a d12) are better or worse. Powerful classes' stat minimums fail to enforce rarity using this, instead boosting the overall numbers for their arrays. While the books also have 'kits' with stat minimums, the arrays don't go as far as to tailoring particular kits e.g. at worst a ranger using the Ranger array only has a 2-in-12 chance of making Giant Killer despite its requirements being fairly modest (Str 15+/Dex 15+). Class-specific arrays are useful for generating NPCs, however.
Methods where you roll a set a numbers, then arrange as desired (IV,V in AD&D 2) are a sort of 'poor man's point buy'). Generally point buy gives more choice and balance. There is some extra individual variation in that different arrays give slight differences in min/max options - e.g. in D&D a 13 for one stat might give a +1, for another no bonus. Or otherwise different breakpoints, etc.
*Basic D&D (e.g. BECMI, Moldvay B/X) rolled 3d6 in order, though after choosing class a character can raise the class 'prime requisite' by 1 by lowering another prime requisite [Str,Wis, Dex, Int; not Dex or Cha] by 2, representing neglecting exercise, studying etc. to focus on their classes' main area.
The Rules Cyclopaedia interestingly has conversion guidelines to AD&D to cover it most often using 4d6 drop lowest: for Basic to Advanced it recommends +2 to a classes' main "prime requisite", +1 all others, while vice versa is -1 prime requisite, -2 all others.

*Mutant Epoch has a straight series of random rolls as the base method, but more interestingly with GM approval a player can choose  to either allocate rolls randomly, or reroll their lowest score. It uses special rolls for some characters (bestial humans [equivalent to Gamma Worlds' Mutant Animals] might use different dice to determine Appearance) in which case these rolls can't be switched, just because its not procedurally possible.

*user Omega on another thread (18/10/14) mentioned a system where 3d6 is rolled for stats, and each is assigned as they're rolled, adding some uncertainty (do you put a highish stat in your most-important stat, or hope you can roll something better?).

*Rolemaster gives characters a current attribute and a "potential attribute" which is the highest it can be raised to with level advancement. Potential attribute is rolled, with a range that decreases as a score approaches a perfect rating.
A slightly similar effect occurs in Powers & Perils, which has a rolled 'base stat' that is multiplied by a purchased multiplier of x1.5 to x4 (bought from a budget of 2d6+14 multiplier points).
 
*one freerpg listed on jhkim's darkshire list (I forget which one) limits one attribute on the basis of another - ST cannot be greater than 4 more than CON (I think) due to the character's frame not being able to support the extra muscle.

*Ars Magica, which has a system where attributes center on an average of 0, IIRC has a system where players pair up to roll and receive opposite numbers (if one gets a +3, the other gets a -3). This is generally a particularly bad idea for game balance, although Ars Magica's 'troupe play' approach does mean that its not strictly one character per player.  

*Spawn of Fashion has random stats, as well as rolls for other things ('body table'), one of which 'hearty' lets a character roll twice for each attribute.

*DCC [Dungeon Crawl Classics] has a 'funnel' system where four randomly generated 0-level PCs are run through a meatgrinder 0-level adventure, with the best survivor promoted to 1st level. Its relative Mutant Crawl Classics also does this, although a significant part of the randomization of a character - the mutation roll - unfortunately (in a weird but balanced way) only appears at 1st level, with 0-level characters not having mutations yet.
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*Dark Fantasy [Tim Barrett's]- has most stats rolled on [4d4]x5. It has partly-random, partly-calculated attributes. The large stats/even number of dice lets calculated attributes be set on the same scale e.g. Mind Power and App are rolled normally while Leadership is 2d4x5 + 1/4 MP + 1/4 MP.

Balanced Random Results:
GURPS free supplement 'Caverns and Creatures' gives characters a random roll on a table, which lists pre-built attribute arrays. Hence stats are randomly determined but in theory no roll is worse than any other roll.
Icons random-rolls stats and # powers, partially balancing this by reducing Determination score if results are high.
An optional idea proposed for D&D is to roll three times (e.g. with 3d6 or 4d6-lowest - potentially the player could choose) with the roll being one number and [25-roll] another number i.e. a character with an 18 will also have a 7. Note this balances in terms of total points but assumes linear costing. For 3.x the total (e.g. 25) should be an odd number as this splits into an even and an odd attribute number, whereas an even number could split even/even or odd/odd (with a lower total modifier).
Another approach to arrays is to roll several dice and group specific numbers as pluses - for instance 10d6 might give 1,2,3,3,4,5,5,6,6,6 = three scores of +1 (single rolls of 1,2 and 4), two of +2 (double-3,double5) and one +3 (triple-6).
A card-based approach that gives balanced approach listed on rpg.net here:
Each player gets a set of twelve cards, from four through nine twice.  
___   ___   ___   ___   ___  ___    
[u]|4|[/u]   [u]|5|[/u]   [u]|6|[/u]   [u]|7|[/u]   [u]|8|[/u]  [u]|9|[/u]      

___   ___   ___   ___   ___  ___    
[u]|4|[/u]   [u]|5|[/u]   [u]|6|[/u]   [u]|7|[/u]   [u]|8|[/u]  [u]|9|[/u]
For each stat, the player draws two cards and adds the numbers together (not replacing them after). Each player will thus end up with six stats in the range from 8-18 and the same number of stat points. (Note that used with 3E-5E, characters can still end up with a different number of odd scores and so differing total modifiers).

Combining Random-roll and Point Buy (on the same character)
A couple of games combine point-buy with random roll. Middle Earth Role Playing (MERP) by ICE rolls character stats randomly (on d100), but gives classes the option to replace one of their rolls, the score allocated to their classes' prime requisite, with a 90. This lets a player drop their lowest dice roll, unless they min/max by placing a roll that's already 91+ in their key score. Gamma World 7th edition (the D&D 4E compatible version) gives the player an 18 and a 16 in the two prime requisites of the characters (or an 20 if both are the same), and randomly rolls the remaining less important scores.
Palladium's Rifts lets characters with terrible physical ability score rolls upgrade them by choosing classes with attribute minimums - Borg especially replaces scores with fixed values, while Crazy and Juicer also have reasonably high minimums. Super powers (Heroes Unlimited, via Conversion Book I) can also replace values. Like the above these act as failsafe systems for ensuring minimum rolls, though they have an in-world justification.

Other combination systems - a houserule from RPGnet for generating D&D stats rolls 2d6 for each, then gives bonus 'rolls' of 6,5,4,3,2,1 between them (one per stat), giving the same overall average as flat 3d6 but some control.
In a sense, random games where a player distributes dice between stats, then rolls them, are also a combination of random and point buy (controlled). Some (Method VI 2nd Edition, 8s for all stats plus 7d6, or the T&T alternative rules method above) roll and then allocate, while others (Amazing Engine) allocate and then roll. The Amazing Engine variant gives a very wide spread of possible scores since both # dice and the final rolls vary and it is more difficult to implement a cap to scores (you can't say "you can't add more dice if it takes you over X" - they just roll, and may get over X), giving its attributes a very nebulous scale.
Another approach for games with separate stats/modifiers is to buy the modifier, use that to calculate the base stat, then roll the remainder, preventing characters from buying stats to optimal breakpoints only (this was suggested in the recent Rolemaster playtest as a houserule - it has 1-100 stats with a breakpoint every 5).

*Blue Planet (Synergy system) has a combined point buy/random roll system designed for an attribute range where 0 is average. The player gets a base # of points depending on the campaign, then opts to spend points to advantage an attribute, or earn points by disadvantaging an attribute; the player rolls 3 d10s with the number of points used as the target number and successes being stat points gained (if advantaged) or lost (if disadvantaged).

*Dragonquest uses point-buy after rolling on a table which determines both # points and maximum attribute buyable (these two things generally oppose each other on the table). While interesting, this has a net effect of generating characters who are across-the-board very competent, or characters who can “max out” an attribute, but who have fewer points to spend to buy these higher attributes. I wouldn't really recommend this since its extremely random - moreso than rolling 3d6 in order for stats - and manages to be uninteresting (providing no input into final character) at the same time.
See page 16 for more discussion of Dragonquest. A similar table shows up in AD&D 2E's Skills & Powers.

Character Modelling & Other Options
A possible third alternative to rolling or using points, Marvel Super Heroes has the option of using "character modelling" to design a character. Here the player assigns numbers to the character (that they think the GM will allow!) and the GM vetoes anything too excessive. This is normally, but not always, done based on a given fictional character; its almost an ad hoc point buy system which works by being balanced against random character that's normally random (if there was a real point buy system, players would probably have to use that). This idea, while abuseable, seems like it could have advantages over many actual point-buy systems; as a D&D-type example, it would allow a GM to do things like weigh up relative importance of attribute values to particular characters (i.e. if the fighter PC wants a high Cha this probably isn't game-breaking since unlike the sorceror it doesn't drive a bunch of spell DCs or damages, so why not let them have it rather than making them dump it for more Strength?). Though in context, MSH doesn't have classes and attributes are generally about the same value from character to character.
This approach is very similar to how GMs in most games will effectively produce an NPC, assuming they aren't diligent enough to stat out every NPC in their city in advance. Apart from MSH it is not often seen for PCs, though.

A few games just gave out pregenerated characters and hence have no system for generating attributes e.g. the archetype systems in Feng Shui (although characters can sometimes adjust stats slightly), Talislanta, or Shadowrun. More extremely, the original Indiana Jones game by TSR, or the recent Marvel Heroic Roleplaying (MHR) Cortex+ game provide not just archetypes but exact characters. On the other hand, the old Demonspawn game books were written for a specific hero, mighty barbarian Fire*Wolf, but did still make you roll stats randomly, possibly giving you a gimpy barbarian.

Timelords (original version) assumes the player will play themselves as a PC, and uses a series of tests/notes to determine the player’s attributes.
(Though you could think of this as a sort of random roll...).

Improving Attributes
After character generation, attributes can sometimes be improved with experience (sometimes not). They may be capped in the same way as initial attributes, or be freely improvable.
Point-based systems like GURPS can let them be bought up with points (sometimes at extra cost compared to purchase with initial points - 3E GURPS doubles the cost after chargen, while in DC Heroes its at x5); systems where xp is spent rather than tracked are also basically this. Level-based games can have automatic bonuses from level (e.g. +1 every 4 levels in 3E), or this can be an option e.g. Savage Worlds (where a character can pick up a stat increase once/rank, also balanced vs. skill raises or new Edges). In either case the bonus is keyed to level advancement, whereas point/xp spending makes a character choose between a stat increase and an increase to something else. Talislanta let characters spend xp to buy either a level or a stat point (25 xp for either), while SenZar has 'fate point' awards for stat increases separate to XP awards. Another interesting system here is Tunnels and Trolls which has very excessive stat raises; stats start at 3d6 but depending on the attribute an increase at level-up can be from + 1/2 level (IQ,CHR, DEX) to +1x (ST, CON) to +2x level (Luck).
Games can also have random checks for improvement (Rolemaster allows checks with increase up to the limit of a roll for potential stat, which is checked upon levelling up and can actually result in losses on a poor roll; old gamebook series 'Where the Shadows Stalk' had a random table which could either improve or worsen stats, rolled at the end of each adventure; Stormbringer has checks as (IIRC) a result of occasional game events; Spawn of Fashan reportedly has a table of attribute increase chances by class, with cumulative chance increasing if the check fails, then resetting to base chance when the check succeeds), or a skill can be learned that increases stats (Palladium). Sometimes stats can also be improved via training (Runequest).
Magic that alters stats can sometimes also be found - this may have separate limits e.g. 3E D&D limits inherent bonuses to +5 maximum (over initial value), while AD&D lets wishes raise scores at 1-point-per-wish to 16, then 10/point from 16-20 (or 11 to go from 18 to 19, with 10% added to exceptional Str each time), and 20 per point beyond (to a hard limit of 25) - but does not record 'initial' scores (making the initial randomization fairer in high-magic campaigns). (The 10 wishes/point is sufficiently tight that a house rule I'd seen was to make that the rate for Limited Wishes'). 2E also has an Enhance spell in Legends & Lore (wizard 8th level) which adds d4 to another's stat (max 22), but drains the wizard [that amount +1] Con points so was rarely used - though if the wizard's Con is <17, they could raise a target's score to 22 and replace those Con points for 1 wish each, exploiting the difference in exchange rate.
Avalon Hill RuneQuest allowed to DEX to be trained to 1.5x its initial value, while for ST/CON the lower of the two could be trained up to match the other score, or both up to SZ if that was higher.
Gamma World has mutations which can increase stats - usually as part of chargen, though characters exposed to radiation could also mutate. GW 4E (1992) used attribute modifiers in an interesting way to calculate amount of increase, with Heightened [Attribute] adding 6+modifier points, i.e. a score of 10 (+0 mod) would gain 6 points to 16, while a score of 18 (+3 mod) would gain only 3 (to 21).
Another mutation "Allurement" has a Mutation Power rating, with the modifier adding to Cha; this is interesting since MP can often be raised, unlike Cha, so the mutation is sort of changing an attribute into a more easily modified 'derived attribute'.
Attribute maximum can sometimes be modified by an Advantage (e.g. Shadowrun 4E has an 'Exceptional Attribute' quality letting a character raise a score +1 above the normal racial limit for 20 build points; stat itself sold separately).

Where stats are rolled randomly and level raises are point-buy you could consider these games to also be a hybrid of random/point buy attribute generation, at least for high-level characters.
A game might have an 'improvement roll' for an attribute e.g. you need to fail the attribute check to gain points in the attribute (Sharp Swords and Sinister Spells) - that randomizes in a way that's counter to the initial randomization of attributes i.e. moves toward balancing it despite it being random.

Losing attributes
On a less positive note, characters can sometimes have attribute loss. Commonly this is due to a result of critical hits or serious damage (e.g. Palladium has this as a side-effect of negative HP (roll on table), Werewolf has 'battlescars' (roll on table), Apocalypse World lets a character 'break' one of their stats (-1) to not die. Ageing or magical mishaps can also lower attributes.
Attribute loss tends to be a feature of old-school or random-roll chargen systems; stat loss in point buy may be a 'fate worse than death'  as a character can be replaced with a freshly built one to avoid the point loss.

Order of Operations
When generating a character, attribute score generation is often (but not always) the initial step.
In games such as AD&D where a random roll limits access to class/race, it has to come first, while in Alternity race is selected first (and establishes minimum scores for point-buy); likewise for Palladium Fantasy where race sets number of dice to roll for stats, race is first. Point-buy systems may put class selection before ability purchase so players know what stats should be prioritized. A system which buys everything (stats, skills, advantages) from a single "pool" might also not have an order of operations, with everything purchased simultaneously.

NOTE: games with random-roll attributes will fairly often roll other things e.g. handedness, class, psionics, educational level, strange medical traits, whatever - this can reduce a need for a complex advantage or feat system since stats already generate imbalance between characters that makes it OK for these extra factors to vary. Point-buy systems may need to give costs for other things besides stats as well.


(thanks to Phillip for additional notes on Dragonquest and Conan; info on systems w/o chargen from J. Arcane, in other threads). 22/1/16 - improvement as a 'mixed' system for high level.

Edit notes: made 'order of operations a heading (*); revised notes on Palladium rolling up (*), MSPE (*), Cop Show (*), "A Little Less Super" (*)

Stat Range
The statistic range (i.e. 3-18, 1-100, etc) normally depends on what dice rolling mechanism is used. Higher scales are generally useful to handle “ability damage” (or random stat determination) but usually also mean that not every point will give a character a benefit. d100 systems can get that fine grained, though recalculating lots of skills may still be painful/discouraged - e.g. HarnMaster, where each skill is the average of three values.
The range of ability scores may just consider player characters or may be designed to consider monsters or other things as well (see Scale, as well as monster design).

Attribute Damage
Depending on the system, attribute damage may be used to handle things like physical damage, fatigue, or spell points (e.g. Tunnels and Trolls uses attribute damage for all three of these), or poison (3.x D&D), though derived scores or non-numerical wounds or conditions can be used instead.
Attribute damage is normally given out using a linear cost; HERO system expresses “Drain” (a power) amounts in terms of purchase points - as the system is point based and its stats vary in purchase price per stat point. Marvel Super Heroes has a mix of damage that is based off stat points (Health off normal attacks, a linear amount), or that causes a Rank reduction (Endurance loss from killing damage) such that losses are scaled up for characters with bigger stats. Dice pool systems are a bit granular for stat loss to work well, but can have proportional losses via rolling [stat] and having successes become the new current stat (e.g. seen in an event card for Arkham Horror). (one example of a dice pool game that does work for stat loss is the Spartacus boardgame - damage can come off multiple stats - see also "take-highest dice pools" post).
Attribute damage is sometimes problematical due to either messy modifier recalculation, or characters being relatively squishy against it in games with heavy HP inflation but relatively fixed stats (e.g. 3.x D&D). AD&D/older D&D tended to not have central rules for ability damage, so individual instances had specific rules e.g. for duration of the effect.
Attribute damage sometimes does not have the full effect e.g. The Fantasy Trip subtracts damage from STR, but always uses the full STR for what weapons a character can wield.
(I think logically a system can have any three of the following four - large enough numbers for ability damage to work, no derived modifiers required, die rolls remaining relevant/bonuses under control, and roll-over).

Other twists: 5E has an unusual case where something similar to attribute 'damage' is used as a check. As a second stage after a primary Int check is failed (d20 + mods as usual), an Intellect Devourer rolls 3d6 and if this roll is greater than Int, the targets Int is reduced to 0.

Modifiers
Attribute scores may be used directly in game mechanics or may have secondary “modifiers” which are used by the game mechanics – compare e.g. GURPS (roll under stat on 3d6) or White Wolf (roll dice equal to the stat) and D&D 3E (Strength score of 13 = +1 modifier on Strength rolls on d20) or perhaps even Savage Worlds' named derived attributes [Toughness = 1/2 Vigour die type +2]. As noted above having a separate rating and modifier may be useful if the raw score is used to absorb damage directly.
The problems with modifiers however are that they a) add another layer of calculations and b) generate breakpoints where extra attribute points don't increase the modifier. This is noticeable for instance in 3E-5E D&D, where odd numbers don't benefit the character. Or DC Heroes has tables where e.g. 5-6, 7-8, 9-10 use the same column (despite 1 AP representing a doubling in score) although optional rules give a slight bonus to an attacker if both are using the same 'column'.
3E D&D attempts to balanced odd numbers by making these into prerequisites for some feats (e.g. 12-13 are +1 and 14-15 are +2, but a character needs a 13+ to learn Dodge or a 15+ to take Two-Weapon Fighting). 4E D&D adds a +1 modifier to all stats at new 'tiers' of 11th and 21st level, so that initial point-buy decisions are flipped at level 11 to 20--all the odd numbers become even and vice-versa).

Some games use both the modifier and the raw score in different cases e.g. 2E AD&D (d20 roll under stat checks vs. combat modifiers) or reputedly 13th Age triggers monster attacks based off natural die rolls - if their natural attack roll on d20 is > target's score a special effect can be triggered.
Sometimes the modifier will adjust rolls for the attacker, while the defender uses the full score as a target number (IIRC Alternity). That could be thought of as a way to cut down the number of calculations modifiers generate - not directly since one is still required for each roll, but by a 'back door approach' since calculations we'd normally forget about, that generate target numbers, are being reduced. Unless modifier is calculated at 1:1 however (e.g. 14 = +d4 on d20) this does generate 'asymmetry' (see post on opposed contests). Also, not every roll is opposed so the modifier is still a bit more work than if a roll was just '+stat'.
At a psychological level people possibly like modifiers since they provide a big attribute score, while having a bonus that's small and so easier to add up.


There are a few models from which modifiers can derive:
 - 1) they can be proportional to the raw attribute; or
 - 2) they may be a plus/minus based on the difference from an 'average' attribute, which gets +0. Building modifiers based around '0 for average' minimizes additional math to some extent, and means multiple modifiers can be applied to a single roll more easily, or that checks with no modifier are more or less balanced. Proportion-based '0 is nonexistent' results means that rules like 'does not add Dex modifier to AC when surprised' are consistently bad for all characters without more explanatory rules. Multipliers to the bonus in a plus/minus system affect weighting of the bonus, while in a proportionality system a reduction is always a penalty. A plus/minus system could make attribute specific costs less awkward (if buying STR costs more than buying APP, say, then 'average' still has cost 0 in both cases).
Certain game functions (weight lifted, damage absorbed) more logically derive from an objective scale where 0 represent nonexistence, while appearance modifiers for instance perhaps suit '0 is average' (i.e. poor looks is a hindrance, high looks is a bonus). If based around +0 for average, the set of numbers giving no modifier is sometimes larger than other bonuses (i.e. 9-12 no modifier, then +1 at 13-15, +2 at 16-18 so that the average range is 4 numbers wide, rather than 3 numbers for each higher bonus level) which might be considered a mathematical flaw. This doesn't always happen e.g. in 3E each 2 points is consistently a 1-pt change in the modifier, and it can be deliberate - e.g. Moldvay Basic uses a modifier scale of -3 for a 3, -2 for a 4 or 5, -1 for a 6-8, 0 for a 9-12, +1 for 13-15, +2 for 16-17, and +3 for 18, which is close to +/-1 = 1 standard deviation, based on the distribution of 3d6.
 - 3) they increase in slowly-widening ranges, so that spending points on a stat gives less and less real return. (Marvel Super Heroes' ranks - these are used on a table rather than being an additive modifier).
A couple of games add in penalties for high attributes e.g. Superbabes gives a HP penalty for high Looks. Nuelow's freerpg "Fairies!" had a personality bonus for high Int but then a penalty for very high, saying that 'mid level geniuses each have their own brand of charm' but assuming high-Int characters would be overly arrogant.
Games sometimes have bonuses for high attributes but no penalties for low attributes - the prime example being Palladium (aside from Rifts Ultimate Edition). The argument for this IIRC being that it encourages role-playing low stats, although it isn't particularly good modelling.

Whether an attribute modifier is applicable to a roll may depend on class, race, or advantages. For example in original D&D (Greyhawk expansion) only fighters could get a to-hit/damage bonus from a high Strength score; in AD&D, only fighters could get a Con bonus to hit points of more than +2 per die; AD&D 2E voadkyn (wood giants, Complete Book of Humanoids) as a racial penalty gained no Str bonus on to-hit rolls (the same could be gained as an artifact curse of 'great strength but great clumsiness'); modern games often have cross-matching e.g. 4E D&D having powers based off various stats, or some 3E feats changing what stat can modify a skill or save (much more possible in 3E as compared to 2E because of the universal stat modifier table).

In some games modifiers are used only during character generation (i.e. Dragon Warriors) and so do not take up character sheet real estate, though break points may still exist. Dragon Warriors is also interesting in having basically two varieties of modifiers; though not named explicitly stats can give a larger 'primary' modifier, or a smaller 'secondary' modifier, depending on how strongly they influence a derived attribute. Most derived attributes in it are affected by one primary and one secondary modifier.
Basic Role Playing/RQ uses a system of category modifiers for groups of skills where a stat might add +1% per point above 10 (primary), or +1% per 2 points (secondary); multiple attributes can affect skills since the modifiers are quite small. HarnMaster determines "skill bases" as an average of 3 attributes, then multiplies these by a *2, *3 or more depending on how easy the skill is to use.

Defining multiple modifiers for a single stat can also be used to reduce the number of breakpoints which have no mechanical effect; it would be conceivable to have a 3E-5E D&D-esque modifier system where every stat point is useful due to having two modifiers, one that increases at 'evens' and the other increasing at 'odds' (on different tasks). (An idea I was toying with for a Savage Worlds hack -  i.e. 2 dice take highest - was  to have attributes give pluses alternating between primary roll and 'wild die' i.e. a +1 score could add +1 only to the skill die, while a +2 would add +1 to either skill die or wild die, then +2/+1, +2/+2, +3/+2. Eventually dropped as too complex.).[/I]
Another idea for avoiding 'dead points' would be to give bonuses to only some skills e.g. with 5 skills, a 10 score might be a +0 to all skills and each point adds +1 to a different skill (so all are +1 at score 15). This works with the concept that an attribute as being really a loose collection of different abilities, e.g. 'Dex' including reaction time, hand/eye coordination and agility; likewise a Cha point might represent 'attractiveness' only if the bonus was assigned to Seduction.
At least one game has two levels of derivation – in effect modifiers from modifiers - and so three columns on the character sheet. In Earthdawn a character attribute generates a “step number” and that “step number” generates a number or type of dice to roll e.g. Str 15 = Step 6 = d10 (the intermediate “step number” can be modified by various factors such as skill levels). It has been suggested the main reason for the multiple layers is to give D&D-like attribute scaling for familiarity to D&D players.
(this sort of effect could be used to scale back the modifier of an attribute e.g. having CON affect Physical skill which then affects Toughness threshold).
 
Attributes vary in their significance/ weighting on tasks - either alone or in combination with other factors. An attribute may absolutely determine success/failure e.g. +12 score may absolutely fail a contest vs. a +13 (Amber, though circumstancial modifiers are common), have a massive effect on success (Tunnels and Trolls stats often range from 10 to 50+, with a random roll of only about +2d6), be rolled as a dice pool, be rolled under directly on d20 as in AD&D (i.e. +5% per stat point), or apply only a minor modifier or bonus percentiles i.e. +1 per 2 points on d20 (D&D), +1 per 5 points on 3d6 (HERO), etc.
Tasks are also affected by skill (or level) to a varying extent, giving rise to variable weightings for both. The weighting may be heavily toward attribute (Warhammer, skill-less systems), 50/50 i.e. 1/2 stat + 1/2 skill (Storyteller, Cyberpunk), or heavily toward skill (i.e. 3.x, BRP). In other cases, stat may have no direct effect on some rolls but may give a cap on skill rating (e.g. Savage Worlds) or additional points to spend; this is discussed in some more detail under Skills.
(thanks to RobMaudib for additional notes).
Note that 'weighting' is affected not just by the theoretical range but also by the distribution within that range. A storyteller character rolls dice equal to Stat+Skill, for example, but stat defaults to 2 whereas skill varies from 0-5, giving more 'spread' in possible numbers and so probably a slightly higher real influence. (Admittedly this is arguable since skill scores of >3 require freebie points or xp).


In some systems attribute and skills do different things making the relative weight hard to determine. Skills may also vary upward more than attribute, e.g. with level, so that impact of attributes gradually decreases.
The 'weighting' can be directly modified in some games, from task to task. The GDW House System (e.g Cadillacs & Dinosaurs) suggests rolling under the 'average' of all applicable skills/attributes; many older systems manage weighting varying from task to task by having multiple sub-systems that work quite differently e.g. a character in 2nd Ed. D&D rolls under Str on d20 for some checks, and applies a smaller modifier for to-hit rolls, rather than having the same bonus for each. A stat/stat modifier will be more valuable to a player obviously if it affects a more critical character function (some as combat), and also varies depending on how easy it is to get modifiers in other ways, or where used by a different subsystem that gets or uses stat bonuses differently (e.g. see damage, post #34). These sort of effects may factor into design decisions about relative cost of attributes, or even how many attributes if an attribute is split because it has become too valuable.
Relative weighting of skill/stat perhaps indirectly affects what can be defined as a stat or skill. For instance, if stat has a relative high weight, a system could have a Size as a stat and then Strength as a skill, whereas a high weighting of skill, or big increases with level, would make it too easy to make characters all heroically muscled or lead to too much level-inflation in Str, suggesting Str as the stat (and hence Size probably an advantage).
Its probably desirable for stats to have a greater impact on raw stat rolls vs. skill rolls (arm-wrestling vs. climbing), though with most mechanics that will generate 'dead space' for the skills- a d100 might allow for +2% per point for one and +1% for the other, but for d20 it probably requires every 2nd point to be unused. Its perhaps surprise D&D didn't decide to do something with this, e.g. to have a 13 Str give a +3 on raw Strength checks.


Modifiers or tables build in a buffer between the score itself and the modifier in the dice roll, giving more leeway in possible stat range (useful for instance to limit bonuses for randomly rolling stats, but a possible problem for point buy since players are incentivized to buy up to specific breakpoints only). For example, HERO uses a 3d6-roll-under system for most attribute checks but characters have a very wide range of possible scores since a check is made by rolling under [8+(Stat/5)]; D&D went from 3-18 using roll under to an open scale in 3.x, with adoption of a d20+modifiers system (which made odd numbers like 13,15 give no extra bonuses; to compensate odd numbers are used as feat prerequisites).
Modifiers most often increase linearly (e.g. +2 to a stat = +1 to the modifier) though certain systems (Rolemaster, FATAL) provide bonuses that start to increase faster toward the far ends of the normal range (Palladium gets a similar effect by increasing the stat at the upper end of the range; a roll of 16-18 initially gets +d6 extra attribute points). AD&D has complex & irregular modifier charts, though it also tends to have bonuses increasily faster at the end of the range (in a random-roll system giving a small chance of a huge modifier).
Some systems while linear cap maximum modifiers i.e. Rifts notes that many bonuses 'max out' at 30 (in the case of PP, strike bonuses stop increases but the characters starts to receive an initiative bonus instead); GURPS caps 'skill defaults' at 20 regardless of the character's actual stat. The capping effect may mean that past this point flat negative modifiers can be ignored (e.g. imagine a rule for Palladium where a character took a -6 Physical Prowess when using their offhand; with a 30 cap for strike bonuses, characters with a 36+ PP would be effectively ambidextrous).
A couple of games include the modifier for a stat of infinity ('Beyond' in Marvel Super Heroes, or '*' in SenZar).

Where a game has fairly limited bonuses on rolls from attribute (and/or used a die with a large linear range), it may require additional bonuses from ad hoc factors to generate logical results. 3.x D&D for example also applies 'size modifiers' to Str rolls to ensure that giants etc. have a reasonable chance of breaking down doors; DMG II for the game has a prodigy NPC template that gives a +2 to one stat, and also adds a +4 on d20 rolls. 3E-derived OGL game FantasyCraft has archetypes like 'Strong' which provide only a +2 to Strength [+1 modifier], but also give a 'double boost' perk (character can spend 2 action dice on Str rolls).
Marvel Heroic gives especially strong characters a Strength dice, but the contribution from Strength itself is low enough that the idea of using it for a 'strength check' is meaningless and questions such as 'can I throw a car at him?' are determined by GM arbitration rather than mechanics; this sort of thing is generally a feature of conflict-based resolution which determines if characters succeed or fail independently of how they attempt to solve the problem. Marvel Heroic characters can however have 'SFX' traits which add to ability-related pools, so that The Hulk can be generally stronger than other characters with the same Strength of 'Godlike /d12'.
Games can also add modifiers on attribute checks routinely for other reasons e.g.:
-4E D&D adds +1/2 level (the 30th level wizard with Strength 10 might have +0 from Strength, +15 from level) for consistency between mechanics.
(similarly, I'd seen an 0D&D houserule where you get to add +level to your (3-18) stat before rolling under with e.g. 3d6, or 4d/5d for a hard check).
-Savage Worlds gives a 'wild die' d6 in addition to the stat die, taking the highest - meaning a wild card with d4 Smarts, theoretically fairly dim, has a better chance of passing a Smarts check than a non-wild-card NPC with average (d6) Smarts; the extra die adds more of a 'curve' to results (reducing odds of critical failure).
-Castles and Crusades has 'primes' which decrease target numbers, largely replacing skills.
These sort of add-ons are potentially undesirable in being a 'kludge' on a system, or at least extra complexity. (Also, arguably players often like big numbers for stats, or at least a feeling their characters stats are superior, so having a low stat + a static bonus for a 'prime' or 'wild die' or 'level' may be less satisfying than a big number despite the same mathematical success chance. Such a system can obscure ridiculous results; a 30th level 4E halfling may be more acceptable with Str 8, extra +15 to Strength checks, than with Str 38). Also note that these improve success modifiers without affecting other functions of the attribute e.g. the Str 8 level 30 halfling has a better chance of breaking down the door than a 1st level 20 Str barbarian, but still has a lower lifting capacity.
Something else reminiscent of this would be 'deadlands' (see 'varying dice type' post): this is particularly weird in that attribute scores are have both a dice type and a number of dice, but only the dice type transfers to skill checks; the number of dice is used only for raw checks so a character might be good at stat checks but bad at skills, or vice versa. In a sense the 'number of dice' is effectively an add-on to the score that kludges the attribute checks but has no impact on skill.

Conversely, 40K future RPGs (Dark Heresy, Deathwatch) has a system for supernatural stats which increases modifiers to normal tasks without giving auto-successes on checks: creatures can have supernatural attributes letting them double their combat modifier, but which doesn't increase the stat for normal d100 under stat 'skill checks'. The bonus increase does exaggerate breakpoints for PCs with unnatural stats, potentially up to +2 on d10 for a 1% difference in stat. See e.g. description here for details of how this sort of thing works.

Games where huge stats provide huge modifiers can also have systems to scale back relative modifiers e.g. JAGS, GURPS. See post 68 (Over the Limit).

Modifiers from attributes are sometimes considered to be independent from the skill/derived attribute they modify, or sometimes no distinction is made between points gained off attribute and other points. 3.x D&D (which opts for the first approach) often uses "skill ranks" (not including modifier) or "base attack bonus" as prerequisites for other abilities. Another instance would be GURPS where skill is bought based off [difference to attribute] rather than final score. Examples of the other approach would be Feng Shui or WEG (d6)Star Wars; in Star Wars a character might have a Strength of 4D, a Swimming skill of 5D, and improvement costs for the ability would be based off the total skill (i.e. 5 character points to raise it to 5D+1), regardless of base attribute.

Modifiers also vary in how pervasive they are through a game system. In some games modifiers may be rarely used and only on a handful of checks (Palladium), in others every task may have one or more modifiers (3E D&D). Modifiers can also sometimes be used negatively (e.g. Size and hiding).

Super Attributes
Superhero/mutant games may also allow for "super attributes". These might just just increase a stat, or be a rule letting characters buy up stats to ridiculous levels [e.g. DC Heroes]; however, some games have separate power rating which applies to "super strength" or the like - this makes power cancelling or powerless alter egos easier to adjudicate. In Aberrant in particular, rolls are [stat]+[mega-stat] which therefore doesn't require lookup of a modifiers table or the like, made reductions for dormancy, or being depowered, or on the other hand bonuses from use of abilities like shapeshifting, very easy to apply. Because 'mega attribute' dice potentially are more valuable than regular stats and make them irrelevant (2 successes per success, or 3 on a 10, plus extra abilities from enhancements) mega-stats can't be purchased to a level higher than the base stat e.g. a Str **** character could also buy mega-str up to **** (but not *****).  This same sort of principle sometimes is used in costing of 'feats' or the like; some Savage Worlds Edges like Charismatic that boost social rolls also have a Spirit minimum, the controlling attribute for these skills, to prevent a low stat from potentially being bypassed with other bonuses
 
Immortal - the Invisible War lets characters buy attribute points (in the game called "motes of immaculum") free-floating, so they can be reassigned between stats ("halo colours") at will. These are more expensive than regular points, although balanced a little bit by Immortal's difficulty system which normally requires additional stat rolls to accomplish actions - i.e. shooting someone through fog with an injury might require a red check (resolve) for the wound, a blue check (awareness) for the fog, and a yellow check (expertise [Dex]) for the attack itself. Without multiple stats being simultaneously applicable on a task, a free floating +1 might as well be +1 to all stats.
 
(relating to the idea of free-floating attributes, rather than super attributes per se, Fireborn has a rule on "stances" where a character can move dice from one stat to another, up to a maximum equal to skill. This handles e.g. a tradeoff between defense and offense, but was useable on most checks. Shifting dice this way on non-combat tasks is also common, the side effects then being largely irrelevant. Shifts from stance are very significant and with 1:1 exchange rate and so make actual attribute allocation in character design maybe not super important, although a characters split in physicals/mentals does also give their initial # of bonus physical skill points/mental skill points, which don't reallocate as easily)

Special Abilities from attributes
Games may also detail special abilities gained from super attributes, e.g. Aberrant added a list of Enhancements with purchase of a "mega-attribute" allowing selection of one enhancement e.g. Lifter (Mega-Strength), Taint Resistance (mega-intelligence), Regeneration (Mega-Stamina), etc.
A few other games with 'super attribute' powers often likewise add specific abilities as well as just a stat boost: Mutant Epoch lets characters with mutations raising stats to roll 'hazard checks' twice; Palladium super attribute powers in Heroes Unlimited often give a grab-bag of miscellaneous abilities as well as a raw stat increase e.g. greater lift capacity, extra skill bonuses, an extra attack for heightened Physical Prowess (Dex), or extra SDC (basically hit points) for heightened Physical Endurance. FantasyCraft has human 'talents' which add a bonus to a stat and related thematic effects - a character with the talent actually doesn't necessarily have a high stat, though it would be common (as point spending is most efficient that way), T&T Deluxe lets characters who roll triples generating stats (on 3d6), add and roll over, and specialists are also permitted to double their die rolls on certain saving throws relating to attributes (for instance, a 'ranger' DEX specialist can double their 2d6 when attacking with a specialty ranged weapon, or add +lvl to other ranged weapons' instead).

Other games have added extraordinary abilities directly based on very high attribute scores. AD&D included spell resistances and Regeneration for scores of 20 and above; the revision of the Immortals rules for Basic D&D included some special abilities for scores up to 100 such as "reload crossbow by hand at double rate" or "can cock trebuchet by himself or herself". Rolemaster also included the option of special abilities for very high scores (a number of lists being provided in Rolemaster Companion III which included the equivalents of talents, spell-like abilities, additional mechanical bonuses like increased healing rates or reduction of specific penalties; Companion IV expanded the list of strength effects to allow use of larger weapons one-handed etc.). While the abilities were often interesting, these often were things other systems would handle in a less ad hoc fashion though rules based off modifiers - Strength requirements for large weapons or a built-in adjustment to healing rate from Con, for example.
In 3.x D&D "Feats" were used to define exceptional abilities, with very high statistics being required for some feats (i.e. 19+ Dex for "Quicker than the Eye", 13+ Str for bastard sword proficiency, or 17+ Dex for Improved Two Weapon Fighting).
 
Conversely, games may also add specific disabilities for very low attribute scores. In Low Fantasy Gaming (d20) characters get bonus languages equal to Int modifier, and are illiterate if their Int mod is negative.
2E D&D notes that characters with an Int of 3 "cannot speak a language, although they may communicate via grunts and gestures", while Amazing Engine characters with a Fitness of less than 10 or so (on a scale up to 100) cannot run.
Games sometimes add these disabilities for situations where a stat is temporarily rather than permanently impaired i.e. Tunnels and Trolls 5th ed., where spellcasting burns Strength points, has a STR reduction to 2 indicatating possible unconscious and collapse (3 is considered fine since character stats are rolled on 3d6 - "while a character with a Str of 3 may have muscles which resemble water, they are obviously a viable character").

Fringe benefits for low scores: games may also attempt to compensate low-rolling players. objectionable game FATAL gives low Int characters a roll for "retard Strength". Some AD&D GMs may give characters ad hoc special abilities to compensate for low stat rolls - see thread here. "Red Box Hack" allows only one PC per party of a given class; if two players want they same class, the lowest total attribute points wins first pick.
Modifiers can also be applied negatively in some cases, and/or it may be beneficial to fail some rolls (whether this is good design being debatable - its often particularly a bad idea with point buy, given that a low score is there a result of high scores elsewhere).

Attribute range: An attributes' range in the context of the system might represent a fairly routine range of ability, or it may be possible to roll or build a character that is severely impaired in an attribute, a setup possibly creating trap options for players. This could happen as a result of overly simplistic design (i.e. the designer sets up a random roll with wonky results, or uses the same modifier for both d20-based skill checks and HP rolls using d6, or decides that 1-10 is an intuitive-seeming numerical range regardless of the probabilities this would generate), or because PCs and NPCs have identical mechanics and a PC gets a more NPC-appropriate result (such as rolling up a 1st-level D&D fighter with 1 HP). Systems can allow maxing out of some attribute, but with the point cost reducing a character to broken elsewhere (Cthulhutech).

Fractional Attribute Values
Attributes may sometimes use fractional values (i.e. if a single whole point is meant to represent a major shift, like a doubling in lifting capacity [suggested by Stormbringer - the poster here, not the RPG]. A couple of games also use fractions in special cases such as D&Ds 18-percentile Strength (where the main score has a score from 01 to 00, generated with percentile dice), or HackMaster (which has percentile values for every stat e.g. 13/47 Dexterity, with characters getting minor increases to each percentage as they level). AD&D percentile Strength is interesting in that bonuses to Strength can be treated in any of three different ways entirely arbitrarily: a single point may skip all the d% business (18 to 19) as with racial bonuses, may give a 10% increase (the Wish, Strength spells) or may improve Strength one 'bracket'/row on the table e.g. from 01-50 to 51-75, or 91-99 to 00.
Pyramid 3-34 suggests a variant rule for GURPS to give fractional ST points to small characters (e.g. Str 2.5) to better differentiate between them, as most otherwise have identical Str (about 2). Allowing Str down to 0.5 also lets the system model birds, bats or 3" little people (there are of course other approaches to do this; see the post on Scale).
FUDGE allows 'half ranks' where each 2nd use of the rating gets a +1 bonus.

Derived Attributes

Many games have “derived attributes” (aka figured attributes, secondary attributes) which are calculated from the basic attributes in some fashion.
These can include
*ratings which are derived from a single attribute in a complex fashion, and perhaps subject to additional modifiers. [e.g. Savage Worlds as noted above generates a number of 'derived attributes' just to convert its dice rolls into averages - so Vigour of d6 becomes Toughness 5 [i.e. 2+half the die], then adds bonuses for armour].
*averages of two attributes (e.g. Runequest/BRP Hit Points = average of CON and Sz; movement rate is derived from STR/DEX in some systems e.g. Mutazoids);
*scores which have a base value derived from an attribute, plus additional points gained from level advancement or class bonuses, and perhaps random rolls (D&D Hit Points; spell points or psionics points in many systems ).

Alpha Omega reportedly has multiple levels of derivation, with normal derived stats (secondary attributes) being used to calculate further derived stats (tertiary attributes).
Dominion commonly uses a number of calculated averages for skills, so as well as skills based on the attributes these averages are listed as "composites" including a Combat Composite (Vigour, Agility), Priestcraft Composite (Stamina, Intuition  - strain on body and soul) and Witchcraft Composite (Intellect, Luck). Whether a characters rounds up or down ("Favourable Rounding") varies - a character can get it if they roll it as an ability, or if initial attribute rolls are low. Across many systems Combat or Magic are frequent situations where multiple attributes get involved also for game balance reasons: a character that's good at fighting or magic tends to dominate the game, other skills less so, though realism can also be a concern (RQ, HarnMaster).

Ouroboros Engine has a sort of 'reverse derived attribute' in Health: Health = ([Prowess+Charisma]/2)
What is interesting about this is that this is justified as being that healthier characters are more charismatic (attractive) rather than the reverse i.e. "the figuring runs in an unintuitive direction".

A derived attribute may be designed to be used in checks of some sort, or to function as a resource (e.g. hit points, spell points) (or both).

Derived attributes can have a result similar to “sub attributes” –discretionary points or special modifiers can give a character a derived attribute quite different to the normal base stat for some purpose, such as characters with low CON but lots of hit points. However, a derived attribute is typically calculated to perform a specific purpose – it will have a scale appropriate to that which is probably different to the base stat, and not interchangeable with it for other purposes, whereas a system using subabilities would probably let a character use their subability for a variety of skills and checks.

Some skill-less systems use derived attributes extensively in place of skill checks. A few examples of this (apart from older versions of D&D) are:
-Dragon Warriors; no skills (only classes in the form of “Professions”): the normal attributes are similar to D&Ds; derived attributes are strongly modified by class/level and include Attack, Defence, Evasion, Magical Attack, Magical Defense, Health Points, Stealth and Perception.
The main writers of DW also wrote a "Blood Sword" game book series: this is relatively complex and has a system like derived attributes (Fighting Prowess, Damage, Psychic Ability, Awareness, Endurance) based off choice of class and level (rank), but without their being any rolled attributes that modify them - e.g. any 3rd rank Warrior has Fighting Prowess 8, damage 1d6+2).
-Gamma World 4E (by which I mean the 1992 Nesmith edition, not the latest version); a limited system ofskills for classes is included, but most tasks rely on derived values including attack (“THAC”), mental attack (“MHAC”), Mental Defense (MD), AC, Health, Perception, Stealth, Remain Unseen, Speed (Base +Dex modifier), and Robot Recognition, and Hit Points. Different classes receive improvements at different rates and most characters get discretionary points every 2nd level as well (a flaw in the system being that 'one point' is added directly whether added to a base-0 value like THAC or suboptimally to a 35-60+ hit point total); the 1-point-improvement gives very conservative increases. GW's system also shows how the derived attribute approach can be slightly clunky, in that lots of numbers are generated, many for defaults that would be 'just an attribute check' in another system. As well as generating confusion or redundancy between derived attribute checks and ability checks e.g. Dex or ranged-THAC to lasso something, Health or Con to avoid food poisoning. Derivation also gives a tighter hold over how numbers are gained, but which may be not particularly worthwhile unless the numbers are very finely tuned to produce results.

-Castles & Crusades uses ability checks for many routine tasks, with skills are replaced by 'primes' - a character choose 2-3 primary stats for which their target numbers are greatly lowered.
Palladium might also be classified as a “derived attribute” system, though a very strange one; it has an extensive skill system but no attribute-check system and (for combat at least, and in a sense attribute checks) largely runs off derived attributes which are based off the skills, instead of using skill checks directly. Characters can get extra SDC (a form of hit points), bonus attacks per round, saving throw bonuses and attack bonuses, as well as attribute increases, by selecting the right skills. Monsters usually have detailed listings of attributes like Str, Dex, Int etc. equivalents, but with this information having usually limited use (there being no defined rules for making attribute checks).
 
Fuzion uses a lot of averages and other figured stats, although its mostly skill-driven.
Characters in derived attribute games are more complex (have more numbers) than in pure attribute games, though not so complex as with skill-driven games. Derived attributes often exist to interface with a particular sub-system, and there can be confusion as to whether the derived stat or the raw stat should apply e.g. Gamma World 4E lists lassoing as a use for a Dexterity check, rather than ranged THACO, or there can be uncertainty about whether to use Health vs. Constitution. AD&D likewise can have circumstances that could call for either a Dexterity roll or a saving throw - each of which would have very different mechanics chances of success chance, depending on character level.
Derived-attribute games may list monsters with just the derived stats, though usually raw attributes are often required as well: 0D&D had monsters with only brief statistics (hit dice, move, etc) and no ability scores, while increasing numbers of optional rules through 2nd edition often meant the GM had to determine monster statistics (Strength for grabs, Dex for trip attacks, mental stats for psionics rolls, etc), and increasingly high PC attributes often left monsters behind, until the High Level Campaigns sourcebook by Skip Williams (and then 3E) included monster stat generation rules. Holmes basic used Dex rolls for initiative, but neglected to include monster DEX scores.
Conversely, Palladium often gives monster ability scores despite these usually having no mechanical effect beyond what would already be included in the combat bonuses.
(See also here in this thread for a description of different types of modifiers).

Usually 'derived attributes' vary more than attributes do, but there are exceptions e.g. 4E D&D, HarnMaster Endurance (see next). Derived attributes bonuses can be kept under control by various methods e.g. using the best of two scores (4E D&D defenses), forcing low scores to be bought up if a modifier is negative (skills and powers skill scores), increasing base value and using a greater divisor to work out a modifier, changing to a different resolution method for relevant checks, etc. [see 'controlling bonus and penalty accumulation'], or making something an average of multiple attributes. A more controlled derived attribute can be deliberately used instead of an attribute to reduce its game influence e.g. Harnmaster originally had a single Endurance score for injury/death checks, but to prevent this being the 'god stat' in later editions made END the average of Str, Will and Stamina. A game might also give characters work-arounds for low values, e.g. advantages /class features letting them substitute another value or otherwise avoid a problem (e.g. the AD&D illusionist has a 'phantom armour' spell which absorbs damage, to go with its awful hit points).
Derived attributes are usually recalculated if the modifying attribute changes, although exceptions exist. D&D "hit points" having a minimum of 1-per-level means a later increase can be better than starting without a penalty. AD&D "Comeliness" is partly rolled, plus a small Cha mod initially, but later increases 1-for-1 if Charisma increases; Comeliness is higher for artificially-enhanced Charisma, though stat increases are so rare in AD&D this may be reasonable. Arduin gives double Con bonus to HPs for rolls over 16 initially, but not if Con is raised later.

A game could theoretically be categorized into one of three groups this way:
*Derived Attribute driven games (derived attributes are used for most checks); as discussed above.
 
*Skill driven games (i.e. most tasks use skill checks); these tend to have lengthy skill lists - basic tasks like Spotting and Listening may require skills (an exception to this being Barbarians of Lemuria, which has only a few abstract 'careers' but which depends almost wholly on GM interpretation). The major problem designing these though, may be balance between skills and the tendency of players to crank up combat skills in particular. Having rules for untrained use of skills is particularly important. (a 'skill driven' game could include both games with level-scaling skills a la 4E/5E D&D, but more commonly skills are improved individually). NPCs in these systems tend to be complex, with lots of skill numbers - it becomes more difficult to fairly generate one 'on the fly'.
 
*Attribute driven games (i.e. most tasks use raw attribute checks)- Tunnels and Trolls, Warhammer Fantasy 1E/2E, Marvel Super Heroes; HERO system, Amazing Engine; Summerland, Apocalypse World. SenZar also used attribute checks for most tasks i.e. skills and saves vs poison and magic etc, although attack and defense are level-based, as are hit points/magic points/attacks per round. Trinity [White Wolf] normally uses Stat+Skill like Storyteller, but has simple 'quick start' rules which has no skills and purely uses attribute checks (Physical, Mental, Social, Willpower [Psi] i.e. seducing someone rolls just Social, hacking a computer rolls just Mental).


Above: the level-up screen from the Crusaders of Khazan video game, showing fairly faithfully the extent of 5E Tunnels and Trolls' level-up options. CON increase is equivalent to getting more HPs, while ST, LK or DEX increase combat ability and IQ or DEX affect magic use.

Attribute-driven systems are usually fairly simple, at least as far as players/gameplay is concerned. This simplicity means that some differences between characters are flattened out -e.g. there's no distinction between a racial bonus to a stat and increase to stats reflecting skill/experience.
Monster statistics blocks usually need to include a full set of statistics (the exception here is Tunnels and Trolls, which often only defines only a single 'Monster Rating' for creatures); this is potentially simpler than other approaches which may require listing stats + a large set of skills or derived attributes, although some derived attribute games don't list attributes at all (older versions of D&D). Attribute-driven games usually have to build in attribute increases to give character improvement - a 'high level' character will have very high STR, CON, etc. and perhaps be somewhat superheroic.
Most of these (apart from early T&T, or the Trinity quick start) have skill lists;, a character either has/doesn't have a skill. To improve a skill a character must raise the related attribute (e.g. a WHFR pickpocket has better pickpocketing than other characters because their career provides DEX raises)- this gives some results similar to e.g. GURPS complex skill defaults but often has other side effects.

This distinction between the categories is related to the split between class/level and skill-driven systems but not quite the same. Most derived attribute games seem to be class/level games but level-based games can also be skill-driven with skill scores set by character level (Rolemaster; 3.x D&D for the most part), or even attribute driven (Tunnels and Trolls prior to 7th edition uses levels, but level advancement just lets the character increase their raw attributes - STR, DEX, CON, etc.). In practice most games are an admixture of the three types, but will trend toward one setup primarily. Where a dual setup is evident, a split seems to most often occur between the combat and noncombat (skill) systems e.g. 2nd Edition D&D has a bolted on skill system (Non Weapon Proficiencies) that are (almost) pure attribute checks, while class features and attacks/saves are derived attributes (notably, it lacks any form of attribute improvement aside from magic so unlike in most attribute-based systems the skills are permanently frozen).
Over time games often seem to evolve toward more complexity, which often means attribute-based or derived-attribute move toward being skill-based. Often an attribute-based game can add in skills giving minor bonuses without roll difficulties having to change much.

Proponents of non-skill based systems may prefer less complexity in character generation as one reason for this.
A game's premise will set some parameters about how much actual differentiation needs to be represented between characters - if everyone is an adventurer, perhaps that's not much. OTOH if a game is really specific, skills can at least differentiate two fairly identical characters based on background. Another question is if gameplay is likely to want or need skills to do extraordinary things or whether they're largely about fire-building or the like - as some said of 2E: "Most nonweapon proficiencies are redundant statements of skills that all characters should obviously have, but are part of the game because the players do not trust the DM to assume the PCs have obvious abilities without some quantitative rule to defend themselves". - M. Keaton on 2E NWPs, Forum, Dragon #257.

(thanks to Rob for BRP notes)

Related Posts: see also skills post for attribute+skill
Edit note 12 July: note I have revised my opinion of 5E, moving it from 'attribute driven' to 'skill based', despite skill bonuses being minor/binary.

Edits: mega-stat/edge purchase notes(*), weighting and distribution (*), low fantasy gaming note

Most fantasy or SF games have a ‘race’ selection step in character generation. In class based systems this gives extra differentiation between members of a class. Individual races may be part of a game for ‘system first’ reasons (the race has abilities that are interesting in the context of the game mechanics), or ‘setting first’ reasons (e.g. a desire to include something different to regular race options of other games). How much of the total 'pie' of character abilities is race-derived and how much is class-derived varies: D&D is maybe 75% class, while in say Tunnels and Trolls race is most important (giving up to x3 or more stat modifiers).

Systems with detailed advantage/disadvantage or power systems may be able to avoid defining a “race” option in chargen completely (e.g. HERO) – to belong to a given race only requires the player choose appropriate powers or Edges/Hindrances. Other point-based systems (e.g. GURPS) allow races as packages of point-costed abilities, sometimes with slight cost modifications compared to purchasing abilities individually. deadEarth is somewhat Gamma World-ish, but instead of mutant animal being a race a character might roll up a single 'cross-mutation' with whatever other animal species, on the standard mutations table; a player can't choose their 'race' and also, your 'race' could change if you're exposed to radiation and gain a cross-mutation, or get a roll that makes you lose a mutation.
 
Systems which evolved without detailed advantage systems usually provide the option of choosing a race, which then gives various advantages/or hindrances. Some of these games (e.g. Pathfinder in accessory rules), have race-creation systems. Savage Worlds Deluxe has a core system that almost could do away with race and just let characters choose advantages, but doesn't quite work since number of permitted advantages/disadvantages per character is fairly small - the optional race creation system adds a couple of new abilities but mainly increases the budget of merits and flaws.
2nd ed. D&D has a system for race creation based off converting monster manual creatures mainly, assigning them stat modifiers and the like but keeping listed abilities and largely not balancing them particularly. A couple of games that are human-only have races corresponding to different human geographical types/cultures (Conan D20, Zenobia). Spycraft attempts to have diversity in characters without it being racial, with the choices being Class and Specialty.

Some races being more powerful than others has been balanced in various ways in RPGs including:
Racial limitations and balancing factors (e.g. being unresurrectable, being unable to choose more powerful classes), point costs, level adjustments (3E D&D), having the race count as an Advantage or feat (e.g. FantasyCraft), having the race count as a class (e.g. Basic), having only a low % chance of being able to play a given race (race determined by a separate roll e.g. Dragonquest, Mutant Epoch), ability score requirements (sometimes of less powerful attributes), needing GM permission to play a member of a race, reduced maximum levels (D&D), slowed advancement or levelling (e.g. 2E AD&D xp penalties, Fantasy Trip overmen stat-buy costs [Space Gamer magazine #28]).

Race & Class
Systems may be designed so that particular races strongly favour particular (hopefully, stereotypical) classes, or can be designed so that combinations are relatively open. Some race/class pairups can be disallowed either specifically, or indirectly as a result of racial ability score, alignment, background or other class prerequisites being unachievable.

Basic D&D allows selection of races as classes (primarily for simplicity). This balances racial options at a (deliberate) cost of flexibility; with this setup it is not necessary to worry about how a given race's stat bonuses or abilities will synergize with class features as the PC can't select another class, and racial abilities can probably be more powerful. Basic races can modify attributes to a variable extent the same way classes do, by lowering Str, Wis, or Int in exchange for raising the "prime requisite" of the race, at a 2:1 exchange rate (2 points lost = 1 prime requisite point gained, in whatever the classes' main stat is). Basic in places found this confining: the Shadow Elves supplement for BECMI included a number of general skills for shadow elves that duplicate thief abilities, so that thief-type characters can exist despite everyone in shadow-elf society being basically fighter-mages (there are also rules to create shamans as a special add-on, costing XP).

Tunnels and Trolls usually allows any race/class though 5E notes that leprechauns 'have to be wizards' since they have innate magic, and also giving them limits in weapons useable (2 dice or less) 'not because of lack of weapons training, but because a leprechaun couldn't use anything bigger than a knife'. Hence, the rules use the class limitation to ad hoc represent a (logical) racial ability. (OTOH, Fairies are even smaller and can theoretically be warriors...they can use very few weapons anyway due to low Str, which would be even more of a handicap to a wizard in 5E T&T due to Str-based spellcasting).
Weapons having a 'dice limit' also produced strange results for an optional Arduin 'thief' class (Sorceror's Apprentice magazine); thieves could use weapons up to 4 dice e.g. normal swords, troll PCs picking up the class were likewise limited to 4 dice although a troll's 'sword' would be about a 12d weapon.

3.x D&D treats unusually powerful races as classes, allowing multi-classing between monster “Hit Dice” and regular class levels; this generally works badly for most multi-class combinations, however (this costs caster levels, means 'first level' (x4) skill points are gained for a monster Hit Die instead of e.g. rogue or ranger, and "level adjustment" may cost an inviable number of hit points).
Dungeon Crawl Classics (DCC) treats races as classes - interestingly, in spite of characters starting at 0-level without class abilities. To qualify as demihuman at level 1, a character must roll the race as part of a 'profession' initially e.g. 'dwarven mushroom farmer'.
Palladium treats some races (the more powerful ones) as 'Racial Character Classes', but most are available with any class. Some 'Rifts' classes are a product of augmentation e.g. juicers or crazies, with some class/race combinations not totally prohibited but with augmentation giving a random chance of death or lobotomization.
D&D variants sometimes attempt to limit particular 'unstereotypical' race/class combinations: they may be banned completely (AD&D), made a higher cost (Aces & Eights, reportedly), or given other penalties i.e. 3E D&D has "favoured classes" to avoid suffering xp penalties, or FantasyCraft (1st printing) had iconic classes/specialties which caused penalties to action dice unless modified via a feat. Conversely (and more difficult) some races may be rewarded for particular class combinations - 13th Age half-elves have an ability to subtract 1 from 'natural' die rolls 1/combat, to synergize with bard/fighter/ranger/sorcerer abilities which are triggered by odd natural die rolls/even natural die rolls.
Classes can also be customized for particular races e.g. if a class gets selections of other abilities, some of these may be race-specific. 3.5 D&D has 'racial substitution levels' which let characters trade in abilities for more racially themed abilities.
As well as thematically-adjusted versions of classes, powerful races might be given 'powered down' versions of classes - e.g. Palladium would have a few alternate versions of some classes for specific races, or 2E D&D has 'demi-paladin' kits giving some paladin abilities to non-human fighter/clerics.
Races may also have 'multi-classing' as a specific ability e.g. the AD&D half-elf (or 3E human, due to having any class as a favoured class). For elves or dwarves, multi-classing may be intended to represent the effects of long lifespans - starting characters being 50 or 100 years old and so get to be trained in two or three classes. A friend's skill-based game had a similar effect by having Willpower modify skill cap (a proportion of total points), with elves having lower Willpower and hence being forced to spread their points around.

Raw stat bonuses can lead to some pairups of race/class being useless; in 4E D&D for instance, slight accuracy bonuses are critical due to monsters of the same level having set defenses (the 'tyranny of accuracy') so that not having a racial +2 is generally considered quite poor. This led to later versions of races being given more choices on where to put their +2. 4e neo-clone '13th Age' tries to fix this by giving out some stat points [+2] for Race and some [another +2] for Class, with it however not being possible to apply both bonuses to the same stat. (Having this work seems like it would depend on how multiple-ability-dependent classes are, and I don't know the system well enough to comment on if it succeeds).  
Another unusual variant is, reportedly, Barbarians of the Aftermath (a post-apocalyptic expansion for Barbarians of Lemuria). Characters here can have racial options like 'mutant bear', but these cost 'career' skill points from the characters normal 4 career skills.

Race Progressions: races sometimes have scaling abilities, either because race is a class and has 'levels', or a race might just have the option to pick extra abilities later, or a race might get abilities scaled automatically based on a skill or something else - quite often an ability might unlock in an ad hoc way at level X (e.g. raptoran flight powers in D&D 3.5), or a weirder example is Newflesh characters for d20 Gamma World which roll a bonus biotech every time their base Fortitude save increases.
Classes can sometimes have race-specific variations that particular races can take, e.g. 3E 'racial substitution features', or some Dungeon World options; this might apply to initial characters, or could be seen as a form of race 'progression' where racially appropriate abilities are delayed to higher levels.
The Pathfinder 2E playtest also has a lot of racial abilities that are feat-based or gained with level progression; this has tended to generate complaints it doesn't make a lot of sense or that 'at 10th level your PF2 elf feels like they're as elfy as a 1st-level PF1 elf'. (A prioritization of game balance over reality). A way to compromise might be to have racial abilities that occur at level-1, but to have actually leveraging them fully require abilities that are gained at higher level, say if an ability requires a difficult skill check to use, or be partly limited like how reach needs 'Combat Reflexes' to use fully in 3E, or be a scaling rather than flat bonus.


Race & Attributes
Race can interact with attributes by providing a bonus to a stat (D&D variants), changing # of dice rolled for stats (Palladium), changing the maximum rating allowed for purchase in a stat without any change in cost (SenZar “GenMax”), applying a separate bonus on attribute checks without changing the score (e.g. Rolemaster adds a numeric bonus, FantasyCraft gives special abilities like 'double boost' to some races, Whitehack and Low Fantasy Gaming have  characters roll 2d20 pick best or pick worst /D&D Next advantage/disadvantage if a stat is racially modified), altering point cost for a given score (LegendQuest), or applying a multiplier to the base attribute (Tunnels and Trolls - normally rounding down, though Deluxe 'trends toward the natural mean', rounding down for multipliers and up for divisors - multiplication as an idea works badly with 'point buy', with 7E's point buy option ignoring race. Multiplying after chargen due to race change can also give crazy stats since bonuses from levelling or magic are then doubled or tripled as well). Adding flat bonuses to a score with point buy and an increasing cost per point basically provides a discount in cost e.g. in 3.5 D&D point buy an elf and a human can both have 16 Dex, but it costs an elf 6 points and a human 10, if the racial mod applies after stat purchase. (The opposite effect apparently occurred in Shadowrun-4E, where it was more optimal cost wise to buy a metatype for cheap Str or Body and then spend the points elsewhere, resulting in mixes like 'troll magician' being most point efficient, presumably due to mod applying before stat purchase).
 
Bonuses can be variable e.g. +d4; even a 'flat' +1 in AD&D can give a very variable effect (none to a lot) due to the strange way the bonus table is constructed, with a boost at around 10 often having little effect while 18 to 19 Strength for example gives a huge boost. Bonuses could be conditional rather than automatic - e.g. "slum dwarves have a 40% chance of gaining a +1 to Str" [hypothetical example], or Mutant characters in the Big Bang Comics RPG roll 3d6 for stats like other characters, but doubles/triples add and roll over. Characters can of course have attribute penalties - often a fixed minus though Palladium for instance uses fewer dice for low stats (e.g. elves roll 2d6 for PE with 12 rolling an extra d6, vs. humans 3d6 with an extra die on 17-18 [differs from later Palladium's 16-18]). Mutant Epoch has pluses for racial advantages but usually gives 'penalties' by defining an alternative rolling method, like d6 Appearance for a mantis-man or 3d6 for a mouse-man's Str,  perhaps since base attribute scores can be enormously variable (10-100) so that it would be easy for a penalty to cripple low-rolling characters without affecting exceptionally high-rolling characters much. Its also possible for a race to set a score at a fixed value e.g. Gamma World 7th Ed. (4E D&D compatible) which had race-as-class, set one racial 'prime requisite' at 16 instead of rolling, or 18 if both races rolled have the same 'prime requisite'.
 
A race might also apply a separate bonus on attribute checks without changing the score (Rolemaster, Castles & Crusades' extra "prime" for humans, 3.5 size bonuses, FantasyCraft 'action boost' powers increasing action die benefits), or an attribute can be modified non-numerically but by noting that for a certain race it operates in a specific context e.g. 'Appearance' stats may be assumed to be specific to the character's race in some systems, or Strength may be relative to other creatures of similar size/scale (as seen for some BECMI Basic D&D races e.g. in the flying gnome supplement "Top Ballista", FUDGE, or "Supernatural Strength" in Palladium). Race can also lead to 'ad hoc' adjustments of task difficulty: 0D&D notes that 'smaller and lighter' characters (e.g. halflings) may be limited to opening open stuck doors on a 1-in-6 chance, rather than the usual 2-in-6; the vagueness gives the GM some leeway to consider the characters Strength on that as well - '...c'mon DM, Bilbulk has a Strength of 16, he should still be 2-in-6...'. AD&D by comparison gave halflings a -1 to Strength but no other open doors penalty, overall a much better chance).
 
Races in some systems can also have "attribute minimums"  either enforcing rarity and/or to give 'plausible' results while keeping stat bonus not too high (e.g. AD&D 2E where no race including giants can have a Str mod. of more than +4). Some character generation methods can also be off limits to certain races e.g. AD&D Unearthed Arcana Method V (noted above under random-roll elaborations) generates very high attribute scores (up to best 3 out of 9d6 for some stats) but is limited to human-only (an early Dragon Magazine expands this to nonhumans).
 
Races can indirectly modify statistics, e.g. by having a point cost to purchase leaving fewer points to buy attributes (GURPS); or similarly 1E Shadowrun required a nonhuman character to reduce points in skills, cyberwear etc. by choosing "Metahuman" as 1st pick on the Priority Grid. Conversely, some 3E D&D races (such as undead or constructs) have "nonabilities" which are inapplicable (-) instead of having a score, which can act as dump stats raising scores overall (the elf with a 6 Con gets a Con of - when it becomes a vampire, avoiding the penalties but keeping high scores elsewhere bought with point buy).
 
To a lesser extent stats can also be modified through different access to classes which add stat bonuses (2nd Ed. Palladium Fantasy). Some systems can encourage PCs with high scores in certain attributes by adding incentives for the player to buy a higher scores (the dwarf gets an extra bonus to saves based on their CON, AD&D, or the Basic dwarf or elf earns bonus XP for a high 'prime requisite'), or removing penalties (the 3.5 half-orc with a 3 Intelligence has PC-minimum Int and so ignores the -2 racial penalty). Racial class limitations in AD&D also slightly discourage demihumans with high scores as many classes with high ability requirements (paladin, 2E specialist wizard, 1E ranger, monk) are human-only.
 
(In general I feel that "racial modifiers" to scores are perhaps largely unnecessary in point-buy systems - they often just lead to complications in the math involved if costs are non-linear (as in 3E). If you randomly roll your stats a +2 Con for being a dwarf makes sense, but if you're going to buy them anyway, you could just buy a higher CON to represent your dwarf being sturdy. The same can apply to classes, too - Maelstrom for instance has point-buy attributes, then adds optional occupation-specific bonuses and penalties that are redundant, largely cancelling out).
(A really extreme case of this that's especially awesome is the deity Ka, in the old Basic D&D Immortal's rules. Immortals normally have ridiculous AC and Hit Points, and can buy fairly silly Str and Con depending on their level; Ka is just described as being as intelligent allosaurus and is like 30-ft-long, which is entirely flavour text in the Immortals' rules; the only power needed to do it is "Size Increase" which actually lets you vary your size, IIRC. He also purchased I think about a 60 Strength, but any high-level Immortal could do the same. Though building younger Ka as a junior immortal would be problematic. The simple Immortals' handbook 'choose a couple of powers' option replaced fairly complicated and messy BECMI rules letting Immortals build avatar bodies from scratch.).
 
Talislanta gives a character exact attributes based on a racial template, with only 1-2 extra points of customization permitted (though multiple templates may exist for one race, which are different classes). In point systems a race may also have a point cost, indirectly reducing attributes (GURPS) or race choice may exist outside the point system (SenZar).

Race Availabilty
Availability of unusual races may be limited by minimum stats, a separate die roll (DragonQuest, World of Synnibarr, DCC where race is built into occupation roll e.g. 'dwarven herder', The Mutant Epoch; this last is interesting in having different tables by level of player experience), or GM fiat (e.g. only one nonhuman per party).
 
Race and skills:
Races may affect skills (in skill-based system, this gives some effects parallel to how race affects class).
Race-specific special abilities that interact with the skill system in an interesting way: reroll X checks (particular stat?), cross-match Y checks, autosuccess for Z checks, particular 'subeffect' gained more easily, action point cost for skill W reduced.

Races often modify skills (both with direct bonuses or indirectly via attribute modifiers, allowed classes). Races can also adjust skill costs (e.g. Dragonquest elves can buy Courtesan ranks for half xp), or cap skills (e.g. AD&D was notorious for giving 'level limits' to demihuman characters). Race can also just give bonus skills (5E D&D) - this makes e.g. all elves 'perceptive', but generally keeps bonuses under control since its redundant with perception from class or background, instead of stacking to make elves better than other characters.
Changeling gives certain 'kith' immunity to botching for about two thematic skills (you can still fail, just not epically); SenZar lets the arboreal Azaar raceclimb trees without a skill check (which 3E D&D also does via Climb speed allowing an automatic take-10), given that without this such characters would logically die.
Star Wars SAGA often gave races particular 'rerolls' on skills, an effect somewhat analogous to how 5E D&D gives some races 'advantage' on particular checks.
(This sort of 'reroll' power is something that is sometimes found as a selectable power e.g. a 'feat' or 'mutation' [like Mutant Epoch's super attribute mutations giving stat rerolls), showing how a race can be built largely of packages of generally-applicable advantages. That being something Whitson Kirk touches on in his 'design patterns' book too, with the darkvision trait).
How race affects skills can flow into how 'class' works, or not, since classes can be design to be skill-driven, or be a different set of rules. An example of this may be Gamma World 5E (the d20-system based Gamma World) synthetic characters, who are specifically bad at multi-classing because they are meant to be designed with particular core programming, paying 'cross-class' cost (2x) for skills gained from multiclassing, despite an increased skill cap.

Races may have specific weapon options either due to size, or via specific racial abilities (free weapon proficiencies, bonuses to damage with particular weapons), or through attribute modifiers (STR/DEX/etc). Note free weapon proficiencies given out for a race synergises negatively with some classes' having the same proficiency (e.g. barring extra rules, elves getting longsword for free is pointless to a warrior who already gets longsword). While size would be expected to modify weapons use, some systems tend to overlook this (e.g. 2E AD&D Dark Sun half-giants have no specified ability to use larger weapons).
Bonus skills for races can be useful in matching them to setting expectations around background e.g. if elves usually come from the forest, bonus Survival (forest) makes sense, or bonus Elven language makes sure all Elves can speak Elf. However, lots of built in skills can also cause problems when a character has an unusual background, say if they were raised by dwarves for whatever reason.

Other racial abilities:
Races can modify advantage purchase costs (e.g. AD&D 2E Skills & Powers trait system elves can purchase Allure more cheaply, while halflings receive a discount on Glibness).(cf. varying point costs). Games with 'feats' may include some race-specific feats. Racial abilities are sometimes defined as advantages - generally, meaning other characters can pick them up as well (though it could also be just to tie them better into some advantage-based subsystem, such as a power with a target number based on # advantages; 3.5 psionics has a a couple of abilities like this, e.g. a HP bonus based off # psionics feats known). This might also be specifically avoided so as to keep races more unique (e.g. 4E D&D specifically gives each race one 'power' unique to it while having a fairly consistent format - elves can reroll attacks ('elven accuracy') while halflings get 'second chance' forcing opponents to miss more often, humans get a bonus class power for their class, half-elves one class power from a different class, dwarves 'resilience' letting them use 'second wind' as a minor action).

Racial disadvantages might be allowed to be "bought off" with advancement if other disadvantages are, e.g. this is allowed for Alternity vampires (Dragon #264) who start with Pallor, and Vulnerabilities (Sunlight, Fire) "at the standard rate of 2x the cost of the flaw".
As well as raw attributes, race can also modify "derived attributes", most frequently hit points (due to size, for example)(see section below); interesting things here include Amazing Engine's Slathorp (which are amorphous and lack 'body points, having more wound points instead), or the way nonhuman anatomies (wings, extra legs) interact with detailed critical hit tables. Systems may also have racially derived saving throws (Battlelords of the 23rd Century). Race frequently also adjusts movement rate, whether that's a stat, derived score or something else.


Race and 'hit points'
Race often affects 'hit points', generally larger characters being tougher (an exception to this is 3E D&D/d20 system, where for creatures built as PCs, stat modifiers or reach would usually give a 'level adjustment' which would reduce total HPs). HPs are sometimes directly set by race, sometimes determined by 'size' (e.g. FantasyCraft). If hit points are just a stat (as in Tunnels and Trolls), race will have to give a stat modifier, which may cause weird effects e.g. a Tunnels and Trolls Troll gets x3 CON score for size, and hence is also much more likely to be able to pass a save to hold its breath underwater (for example). Using 'race as class', race may give class variations e.g. in Hit Dice. Races may get an ad hoc modifier, e.g. half-giants in AD&D Dark Sun doubled their Hit Dice (e.g. a fighter would get 2d10 per level). Extra toughness can also be generated by special abilities, such as a 4E dwarf having 'second wind' as a free action.

Final Notes
As with advantages, the more complex and detailed the games' other systems, the more specific a character's racial abilities have to be e.g. compare GURPS where exact Strength, movement, advantages and disadvantages would need to be defined, vs. say FATE where a character could have an aspect of 'Worlds's Greatest Donkey Mathematician' and apply the aspect bonus as required to running (physique rolls), not being allowed in bars (a compel), etc, albeit with GM adjudication frequently necessary.
Racial modifications are generally similar regardless of attribute systems, although this might be expressed as an attribute modifier in one game and miscellaneously in another. For instance, Rolemaster notes that different races have strengths in different areas of stat bonus modification (elves), background options (humans), Resistance Roll modifications, or Body Development ranks; comparatively a different game might just have various races getting bonuses to different attributes and yet cover almost the same ground due to the scope of what attribute scores cover.

Balancing racial special abilities: special abilities are sometimes 'overly limited' in game terms of what they can do e.g. 4E D&D thri-kreen have 4 arms - which just gives them the ability to draw objects as a free action (=equivalent to the quick draw feat) - and an encounter power with multiple claw attacks.
Special abilities might sometimes need extra skill points invested to work (e.g. making mind flayers take 'martial arts' skill ranks to get brain extraction special ability to work).

Racial subvariants
Fantasy games can have multiple slightly-different versions of races.
3E D&D for instance has 'jungle dwarves', 'desert elves', etc. as options in its 'Unearthed Arcana'; the variant choice is available for free and lets a player substitute a couple of racial abilities for e.g. terrain-based abilities. Some of these alter ability modifiers, etc.
AD&D 2E in contrast sometimes treats a racial variant as a 'kit' e.g. jungle dwarves have a  'barbarian, jungle dwarf' kit in Demihumans of the Realms. Making it a kit limits class choices in this case; unlike other race options it also takes up the kit 'slot' (i.e prevents a character being a Champion or Verminslayer or whatever).
In other games, FantasyCraft often has 'splinter races' which require a Feat to select. Talislanta has individual archetypes for racial variants, incorporating specific skills / base attributes /special abilities (there is one archetype, gao sea rogue [pirate], which is complicated by allowing various races). Some races are explicitly barred from picking up magic as a secondary skill where this would be imbalanced - the warrior race of Thralls are racially unable to understand magic, while the dual-brained 'Sindarin' normally have huge Int scores, but have one brain go crazy and die should they try to learn spells.

Race customization: some games may allow characters to slightly vary or tweak racial abilities. One of the more extreme cases was 2nd Ed. Skills and Powers, which gave a race a budget of character points which let them buy either the normal abilities (plus very slightly more), or adjusted abilities including abilities from other variant races and slightly specific or unusual abilities - extra weapon bonuses for instance. It was slightly weird in that you could, for example, have some elves who didn't have infravision or other really standard abilities. This sort of thing let players duplicate some extra abilities that might often be 'feats' or 'advantages' like 'racial weapon familiarity' within the expanded race-creation system, or could also be used as a race-creation tool for variant races by the GM if restricted from player use. This sort of thing might be duplicated by DIY race-creation in other games like Pathfinder, with GM approval, if you didn't have enough options already. Advantages/disadvantages can also modify - for instance it might be useful to have a disadvantage that takes away a racial ability for characters with unusual upbringing (e.g. elves who were raised by humans and can't speak elven) or ancestry; disadvantages are probably uncommon enough that's more reasonable than say Skills & Powers.

Interspecies Hybrids
The  2014 playtest revision of Synnibarr uses points to buy races, and lets characters buy multiple racial packages to be a 'hybrid' which I thought was interesting - the hole here being that humans cost 0 and get some minor abilities. Starcluster 3E has a system where a hybrid rolls for inheritance of individual traits/body parts; other games may have defined races for each racial type (like D&D half-elves and tieflings). 3E D&D has 'templates' which add sets of bonuses for being e.g. half-dragon, half-fiend, etc. Systems where race is represented by buying individual features avoid a need for cross-breeding rules - a character can buy a few appropriate abilities and claim to be a cross-breed. If a game has a race creation system (Pathfinder) this might also be used to build half-breeds.
A d20 game in progress here has Races which "consist of two Features each; a Primary and a Secondary feature. You can create a hybrid of two races by taking the Secondary Features of two different races."
Class systems sometimes include classes that progressively mutate a character into a different race, e.g. Dragon Disciple in 3.5 D&D (or Stoneblessed, or Gamma World D20's 'hybrid' advanced class [i.e. prestige class]). This approach mostly suggests a 'target' race duplicated has to be more powerful than the base races available, since otherwise a player would just choose to be the target race to begin with.
TrollZine #3 has an inheritance table for Tunnels and Trolls which lists 'Half Breed' as an option - "Pick an attribute multiplier of another race for one attribute and substitute for yours. You resemble that race" - the option here having to be an attribute modifier as T&T races only give attribute modifications. Unfortunately not selectable except at random.
Whitehack as noted in attributes, treats a race as a tag which counts as one of a characters' 'group' choices, and gives 5E-style advantage on two attributes (roll twice and take the best), chosen by the player rather than expressly defined AFAIK, as well as disadvantage whenever the GM deems that appropriate. Half-breeds pick only one attribute, a bit suboptimal although the GM may also choose to ad hoc penalize them less.

A common issue with half-breed races, for some reason, is that they may get abilities belonging to neither parent race.

Systems without races
A couple of class-based games that don't have a race selection option, instead add another 'step' to maintain a satisfactory number of choices. Mostly this means human-only worlds:
*the d20 Conan RPG has different races that are human ethnicities, but in a D&Desque format including attribute modifiers, skill bonuses, weapon proficiencies etc.
*Spycraft has every character select a 'talent' and a 'specialty' (FantasyCraft inherits this, with humans only getting talents but specialties for all, for an unusually complex chargen system). Talent is more race-like (attribute modifiers and special abilities) while specialty resembles a background (in the 5E D&D sense) or sub-class/kit add on to class.
*Beyond the Supernatural (Palladium) has a two-step class selection setup, where a character chooses a "Psychic Character Class" (P.C.C.) that includes attribute modifiers and special powers, then an "Occupation" which determines their skills.

Race and Game System
In games where there are several different games that are highly compatible, a character's "type" (which you could think of as either race or class, maybe) is actually determined by which game system you're using.
For example, its possible to have a Storyteller game where one character is a Vampire, one a Mage and one a Wraith, though those are technically different, compatible, 'systems' rather than being a 'pick race' option that's explicitly outlined in chargen. Something similar might be said to exist for Palladium (e.g. Rifts), or in other games where crossovers are allowed e.g. a D&D/Gamma World hybrid game might have what's basically an informal  'pick world of origin' chargen step for the player, before the normal race and class selections.

Race and edition changes
A race can have a 'niche' in one version of a game and have it eroded in other versions. One example of this is the AD&D gnome, which seems to exist to let dwarves be magic-users; another example is the Pure Strain Human in some gamma world versions - having a niche in that with more Int they have more technology, vs. this becoming e.g. a 'class' function (Examiner) in 4E (1992 edition) when classes were added.

These are defined traits which (unlike attributes) not every character has.
They might be binary (you have it or you don't), or come in multiple levels (less commonly).
Usually cost is related to how useful an advantage is, although apparently GURPS 1E (thanks Estar) sometimes under/over costed advantages to enforce rarity instead -http://forums.sjgames.com/showpost.php?p=1566965&postcount=3
Some advantages are fairly specific (you get a +x in a particular circumstance). Others are not just bought but then must be 'defined' further e.g. a 'noble background' would have a rank and social obligations that need to be described, or a Leadership feat cohort or 'mentor' might need to be built as a separate character. Too many advantages needing detailed specification can become wasteful of GM time; sometimes advantages might be designed to be interpreted more 'on the fly' to avoid this (maybe Fate Aspects).

Sometimes an advantage has an opposite 'disadvantage' - these might be costed the same (+10 for Precise Touch, -10 for Butterfingers) or a cost system might (as in the case of GURPS) reduce the points gained for a disadvantage on the grounds that a character is only likely to take one that doesn't have too much impact on their specialty/concept. Character points are sometimes related to some measure, such as 1 CP equaling a week of training. More likely costs will be related to utility of an ability, although some systems also set an artificially high cost for certain things to keep them rare (e.g. GURPS "Unusual Background" advantage).

A disadvantage system (as pointed out here) has the goals of compensating characters for flaws (which they might've taken for RP purposes anyway), and encouraging characters to be more interesting. These two goals are slightly at odds with each other since the second requires a PC be encouraged to load up on disadvantages up to a limit, which is more likely if they aren't actually that much of a hindrance, while the first suggests cost should be allocated fairly.
Systems while have extensive 'super power' subsystems may avoid needing distinct advantages since most effects can be modelled using powers (e.g. Marvel Super Heroes), though more commonly there are two distinct subsystems.
Binary-cost advantages which require checks may piggyback off another rating for use - skill, stat or another check - thus making the other rating slightly more valuable to that character. They can also just add bonuses to other checks in some cases. An interesting sort of example of the possibilities of design here might be the 'Wealthy Relative' advantage in DC Heroes 3E - the character gets a wealthy relative to sponge off who has a Wealth rating of (PC's Wealth +3) if they can be persuaded to part with their cash. A similar effect could be modelled by letting a character just buy a Wealth rating with a cost multiplier to account for the limitations here - parallel to how it modifies power 'factor costs'. (As designed, the advantage has a cost flaw in that there's a Wealth minimum before the advantage is cheaper than just buying a Wealth 3 points higher; this is a general problem with advantages that are fixed-cost, fixed benefit when ratings have nonlinear costs).  

Older game systems can also have "disadvantages" and "advantages" that function in an ad hoc fashion - without their being a single, unified way of choosing them, or costing method. For instance, Palladium lets some mutants buy up/down mutant features ("Speech, Looks, Bipedalism none/partial/full) using BIO-E, plus has random background tables, insanity tables, % random psionics, and other specific rules governing edge cases for other classes and races: a mutant could roll 'ambidexterity' as a mutation abnormality, or a phobia of the insanity tables, but they're not generally buyable. A "cyborg soldier" (Ninjas and Superspies) can 'sell off' body parts for more bionics $ budget, but most characters can't (the way a GURPS character could). A 2nd Ed. D&D wizard can trade spell schools to 'specialize', something that could've been (but wasn't) a feat in 3E. Talislanta has no 'advantage/disadvantage' system, but has a huge list of archetypes that are very specific - these may include a number of abilities that would ordinarily be separate options as built-in features.
(It is also quite possible, of course, for a player to choose to have problems or flaws without there being explicit mechanics for this at all. In a sense disadvantage systems distrust players to not naturally do this.)

Disadvantages usually have an up-front benefit i.e. extra character points are gained for taking a disadvantage, such as in GURPS. In some games, e.g. Savage Worlds, the character just selects an additional advantage for each disadvantage or equivalent other benefit (Savage Worlds does however have more powerful advantages which just have higher prerequisites, like 3.x D&D feats do). Other systems have more complex costings - GURPS has variable costs for everything; MasterBook divides advantages and disadvantages into multiple levels, with a disadvantage allowing another advantage of the same level but not usually one of a different level, so there is no exact comparison between levels possible. HERO distinguishes between fixed-cost advantages/disadvantages (“adders”) and those that apply proportional cost increases/decreases, with multiple disadvantage ratings added to give a cost divisor; DC Heroes (3E) uses a “factor costs” table (i.e. a factor cost 5 skill/power with a +2 cost advantage would become factor cost 7, and would use the "7" column to determine total point cost for N points of a skill or power). (See in the powers section here for more on advantages/disadvantages for Powers specifically).
As noted under attributes, attribute points might be purchased as a separate step to advantages/disadvantages (typically before), or there may be a single 'pool' used to buy everything. The second approach means that a disadvantage can potentially generate attribute points (not just advantage points, or skill points), or that a character buying an advantage is potentially giving up attribute points to do so. Using the same pool therefore makes advantages more measureable in terms of what they're worth, and hence the added flexibility comes with need for more detailed analysis of costs, to avoid min/maxing problems. 'Young' in Savage Worlds for example, comes with the problem a character gets one stat point for it (as a Major Hindrance) -but it has an effect of immediate loss of two stat points. Or SenZar has advantages like 'attractive' or 'keen senses' that add +1 to stat per level, which often cost the same as adding +1 to the stat, for stats less than 11. Segregated subsystems would make this sort of cost comparison impossible or at least indirect. Likewise advantages and skills being bought with the same points makes advantages giving '+x to a skill' potentially broken cost-wise (as seen with GURPS Eidetic memory, but also an issue with advantages that let characters apply skills to duplicate other skills, as either the advantage or the other skill may be overcosted).

Instead of gaining or costing points initially nWoD Storyteller and FATE instead provide xps or 'Fate points' when a disadvantage comes up in play; in FATE this allows the same traits (Aspects) to function as both advantages and disadvantages - although being freeform (instead of from a defined list) they are limited to only adding a bonus or penalty, instead of having defined effects.
Aspects in FATE are generally very open to interpretation- a few examples may be given but unlike e.g. GURPS, they don't need detailed mechanical differentiation (some FATE variants have 'stunts' however, which may have their own little subsystems). On the other hand as it is said here: Aspects can easily be bland; when "Trained by Swordmaster Vortex," "Strong," "I'll do anything to help my true love," and "Woosh!" all have the same mechanical effects, it's pretty easy to see why some people might think the mechanics all feel samey. (Anyone complaining that Savage Worlds is 'bland' should go check out FATE).
Aspects do seem to go beyond what normal advantage/disadvantage systems do, being integral to or replacing a lot of game systems in some Fate versions (e.g. Legends of Anglerre). They are sometimes criticized for requiring 'fate points' to be spent for bonuses the GM would often just give for roleplaying in a traditional system (e.g. post by Obeeron here).
Miscellaneous WTF: rpg.net thread on 'negative compels' (aspect compellable due to being inapplicable) here. Aspects being 'paid for' anyway perhaps may also encourage strange aspects to be allowably invoked, GM willing e.g. a player might be able to invoke a weird aspects since its essentially a 'reskinning' of just invoking another aspect they could have used instead. That could give rise to weird narrative scenes (like where in Mad Max: Fury Road a sudden hallucination makes Max block an arrow to the face just in time - which you could see happening in FATE easily).

In most games how advantages/disadvantages can be tested with checks or interact with dice probabilities is fairly nebulous compared to, say, the standardized bonus systems for how attributes or skills give roll modifiers. GURPS has a '+1 per level of trait', while Cortex as noted above uses die codes (d2, d4,d6, etc.), and Fate has a blanket 'Aspect gives a +2' rule (although its usually fuzzier than that, perhaps since aspects are typically special traits that are often more qualitative than quantitative). Systems with detailed predefined advantage lists sometimes are built with a general guideline ( e.g. +4 for D&D feats, +2 for a Savage Worlds edge), but with individual exceptions where a trait is more valuable.

Other Elaborations
*World of Terath has an interesting disadvantage costing system that involves step dice. Disadvantages are rated from d4 to d20 (rarely d100) rated in severity; a rating in it is assigned (giving bonus points) and in game a roll against the rating with the appropriate die indicates that the disadvantage applies. Hence the more debilitating the disadvantage, the more points are gained from maxing it out. 'Deep Sleeper' is a d10 disadvantage and could earn up to 10 points (the actual # being rolled against to wake up), while illiteracy is only d4 and so worth a maximum of 4 points (roll against actual rating to read something). Some disadvantages also had variable severity and could have various die types.

*Gamma World 3E has defect mutations (like inability to feel pain, a need to be immersed in water occasionally, etc) - these have a 'defect score' rolled on 3d6, similar to an attribute score (or mutant power).

*Instead of there being a general points pool or having each advantage chosen balanced by a disadvantage, advantages/disadvantages could just cost attribute points. e.g. 5E D&D lets humans trade stat points for a bonus 'feat' (and later, feats replace ability increases). 'Milleniums and Mutations' supplement for T&T lets mutant animals pick up to 4 special abilities, but with each costing 25% of an attribute score (rolled randomly; additional rolls of the same score are ignored). Attributes are usually fine-grained enough that this would allow multiple degrees of advantage/disadvantage. This setup makes the relative cost of advantages/disadvantages very measureable as noted.

*ROAR (Rick’s Own Adventure Rules; a free rpg, now extinct) did not have a separate purchase of advantages; instead attributes provided pools of “faculty” points if high, or “flaw points” if low, which had to be spent on appropriate advantages/disadvantages. A high DEX character might be ambidextrous, while a low CON character might be a haemophiliac or sickly; a low CHA character might have social issues, pox scars, etc.

*Synnibarr 'skill points' (which can actually be used for a variety of things e.g. purchasing mutations and bionics) are based off [2x total of initial stats] rather than being a fixed pool separate to stats, or bought from the same pool as stats. The issues here are basically that a character is doubly boned by low rolls - a low stat and less skill points - and its questionable realism as well in this implementation (higher STR = more knowledge skills, or mutations?). Characters can get bonus skill points for choosing defect mutations, and also get bonus skill points by levelling, with more powerful classes not getting bonus skill points from level until higher levels in some cases.

*SenZar has a point system used to purchase advantages and attributes (from a budget of 100 points), which unlike say GURPS is wholly separate to its experience system. Characters can earn points for good playing, while XP from killing monsters adds to level (which has basically no effect on attributes or advantages). In essence it works as GURPS + D&D running side by side, instead of the usual hybrid where character level gives more points. AD&D 2E with 'skills and powers' is distantly related in that characters could buy a number of traits as well as having a class/level, but didn't combine the attributes and traits budget or allow traits to improve after character generation.

*Random systems may have random disadvantage determination i.e. HarnMaster characters can gain bonus attribute points by rolling for defects on the Medical Conditions Table, with some races trading out specific results e.g. elves gaining sterility instead of pox scars.
Gamma World (older versions) has mutation tables which includes both positive and “defect” mutations.
Mutazoids has random-roll mutation defects, but characters can spend points (=Power attribute) to buy off some of these, instead of purchasing powers or general mutations. (Mutazoids is also interesting in that it rolls descriptive mutant features area-by-area; a character rolls on a specific table for each of head shape, nose, mouth, teeth, tounge (sic), ears, eyes, arms, hands, torso, legs, feet, hair, skin texture, body colour, reproduction, diet, markings, head ornament, tail and wings. This does prevent characters rolling duplicates, but is a bit slow given that most rolls are 'normal' and can give mild-mutants that roll badly and get severe mutations.
 
*Unisystem has a number of advantages that are packages of minor advantages e.g. a Slayer in the Buffy game gets some levels of 'hard to kill' and stat bonuses, as well as enemies and obligations; an "Occult Investigator" in Army of Darkness gets a skill bonus and mental attribute points, as well as a drawback of curious about occult things. It does this sort of pre-packaging in an apparently ad hoc way, as compared to say GURPS where packages (like races) are bought as packages.

*Pathfinder has two separate advantage systems, Feats inherited from 3.x and an optional system of "traits" which fill a niche for lower-powered abilities (since feats are large/granular and there's no varying costs as in for example GURPS).

*3.0 feats could arguably have had a role in replacing subclasses/multiclassing, but ends up being redundant with 'open' multiclassing.

*DC Heroes 3E is also interesting in that it modifies bonus points for disadvantages when taken by a character with a higher point total. If a character is built with x2 normal starting points, a 25-pt (base) disadvantage would be worth 50 points, while a Superman-type built with x10 normal starting budget would get 250 points for it. Hence RP-based disadvantages that are always somewhat of a nuisance (like Married or Secret Identity) don't become less attractive as the number of points splashing around in chargen goes up. 3E has a specific problem with this and the "Pet" advantage, which should be exempt from scaling IMHO as the pet already has an inherently variable cost based on how awesome the pet is (1/2 what it would cost to build as a character).
A (sort of) similar idea is found in the Invulnerable RPG, where characters of higher Awesomeness Level can take more points in disadvantages [8+2/level], whereas GURPS has a GM-specified limit that may be some multiplier of a characters' total initial points. In these cases characters can take more disadvantages, but the individual disadvantages are still as bad. Some might be partly offset by some skill or power that becomes more affordable, however.
Some  classes of disadvantages relating to powers, alternate forms, etc. may be limited to superhuman characters (for instance, BESM notes this is the case for some of its disadvantages) - if a character is limited to a maximum number per category, this as a result indirectly increases the allowed size of the 'disadvantage pool' for supers.

*HackMaster has random-roll disadvantages but with results each providing different numbers of 'build points' (BPs) based on severity, or lets players 'cherry pick' disadvantages for set (lower) numbers of BPs. It lets BPs be spent to reroll some random rolls as part of chargen (e.g. ambidexterity, family heritage, social status).

*HERO I think allows for advantages to have separate modifiers themselves like limitations that will modify their cost. Adding a 'requires a roll' modifier for instance could create a sort of lower-value 'partial advantage' that works sometimes, instead of reliably.

*Advantages that require skills to use: Mutant Bikers of the Atomic Wasteland [2nd Ed] has "Gifts" (including powers), and then has skills to use those, which default to Good instead of having to be bought up with skill points (a "Good" rating normally costs 3 points out of 30 initially; so the intention is to make it possible to roll against Gifts without these actually handicapping skills separately, the choice of the Gift itself being the resource). Similarly, Savage Worlds powers (such as magic) have a skill-rating bought separately; this does default to untrained and so costs skill points in addition to the Edge to get the power.

*The 'Golden Open Gaming' D20 system variant expanded the basic D20 stats by including feats which allowed characters to roll up additional attributes (Appearance, Sanity/Insanity, Luck, Education, and so on). Unfortunately while having the attribute was often a prerequisite for other feats (like mechanical skills requiring Education, or seduction feats requiring Appearance), paying the feat and then rolling often generated scores that were less-than-average that shouldn't logically meet prerequisites. Hence a character might pay a feat to be ugly due to a poor roll (e.g. rolling "Appearance" with 3d6 and getting a 5), but could still select the 'stunning cleavage' feat based off Appearance. There was also a "Limited Education" negative feat which was bought after the Education feat, limiting maximum to 8 - 'so your salt-of-the-earth mechanic type can purchase assorted technical feats without becoming a brainiac' - but with a negative feat granting a bonus feat, taking this made Education actually free. This sort of mechanic could work if the game had also had a 'disadvantage limit' on how many disadvantages could be taken - if so while Education was still free with that disadd, the disadvantage would at least be taking up a 'slot' that could be used to take another disadvantage that could fund any feat, thus being an (albeit minor) limit upon character-design.

*Tristram Evans' *Venture Bros, has a random-roll system where a character can refuse a randomly-rolled trait, but then has to roll 2 random traits instead, giving some player buy out if they especially don't want some character feature but at a cost.

*Low Life for Savage Worlds has a weird race (Bodul) which has a 'strange' racial ability letting them choose as many hindrances as they want (getting an edge for each) instead of the normal 1 major/2 minor. Low Life hindrances tend to include physical weirdness traits like no arms, making it work as a mutation system particularly.

*As noted earlier advantages can be purely either/or, or come in multiple 'levels'. D&D 'feats' sometimes automatically level up, or sometimes a player needs to select another feat to improve them. 13th Age lists specific tier-based benefits for each feat ( a character automatically improves their specific niche rather than being able to shift their focus).

Psychological defects may include absolute limitations (“the character can’t do something/must do something”), trait values that are rolled against (i.e. SenZar has “karmic saves” i.e. roll over your arrogance score to resist being taunted), or may use an attribute check such as a Willpower check to avoid disadvantage effects (this last may be abuseable). Disadvantages with their own trait scores may overlap “Personality traits” systems found in games such as Pendragon. Similarly, Dying Earth builds mental disadvantages into characters automatically by assuming characters are susceptible to most forms of temptation (requiring a roll), unless specific resistances are purchased to traits i.e. "resist Gourmandry" or "resist Pettifoggery". Unisystem lets characters buy off RP-based disadds with good roleplaying, rather than just spending points. A game may sometimes sell a disadvantage for having to play a character in a derpy fashion separate to actual INT score e.g. SenZar's disadvantage "Total Stupidity" (which characters with high Int could actually take - meaning they're just especially thoughtless). (This is handy in SenZar for giving PCs some human foibles, as its relatively cheap to put up every stat to human-max at at chargen).
 

Scope of Advantage/Disadvantage systems
The scope that advantages and disadvantages covers depends a great deal on how the other main components of a character - attributes, race, class, and skills - are defined e.g. we can look at  
*advantages vs. attributes:
*advantages vs. skills
An extreme case is the PDQ system: characters have 'qualities' which are advantage-like, although they vary in rating rather than being yes/no; these can be skills, racial features, etc. A quality is rated -2 to +6; they can be anything but are 'spun' in that direction. If a character is "Hook-Handed" at +2 it can be used as a weapon or threat or etc, (unlike FATE) only if rated negative does it generates complications.


-Advantages vs. Attributes:
*advantages vs. attributes: An advantage in one system might be an attribute in another - for example compare the 'Handsome' advantage in GURPS vs. Physical Beauty in Palladium (a statistic which can be randomly rolled). Conversely, Palladium psionics are rolled off a table as "none; minor; major" and then powers provide ad hoc bonuses, whereas another system might have a Psionics attribute (e.g. Space Master), and the modifier provided by a power (e.g. Dodge bonus from a Danger Sense type power) might be based off the Psi attribute modifier. FATE variants often only have 'Aspects' and no attributes. Special abilities such as 'ambidexterity' might be advantages in one system (perhaps with a stat requirement to purchase), but occur automatically off a high attribute in another system that doesn't use advantages.
In a point system, Advantages/Disadvantages may assist a system in reducing its number of attributes and/or keeping its attributes balanced e.g. GURPS treats various levels of “Appearance” and “Will” as advantages, leaving it with only 4 core stats. Unisystem has advantages which often add bonuses directly to stats (despite these being rated only 1-5).

Advantages can also overlap the same sort of ground as 'subabilities', though often defined advantages will give larger bonuses than are given by attribute and/or stack with attribute bonuses. There can be problems with overlap between advantages/attributes if one is cheaper and better - e.g. Champions (some versions) selling off END to buy 'End Reserve' instead for half the cost.
Disadvantages can be useful in e.g. supers games for adding limitations /weaknesses to characters - a human element - even though the system might be set up so that heroes have stats higher than human normal in every way (DC Heroes).
Statting a game element as an 'advantage' vs. an attribute perhaps affords a little bit more flexibility with it - in some games advantages can be further modified with other enhancers or situational limiters. Multiple options might be built in more easily than with attributes (at least, ad hoc defined attribute effects are relatively rare) - for instance HarnMaster has a Voice attribute which specifically improves Singing and Persuasion type skills only, whereas a Deadlands character could choose 'The Voice' edge and take one or more of a 'soothing voice' (Persuasion bonus), 'threatening voice' (Overawe), or 'grating voice' (Ridicule). Advantages/disadvantages being descriptive instead of just numbers sometimes means creating extra rules ("a character with Monstrous Appearance cannot also have Attractive" for instance, as opposed to different numbers which are mutually exclusive - a score can't be -1 and +1 at the same time).
Potentially if a factor affects another attribute - such as Size increasing Strength - probably buying of both will work better if handled at separate steps, although it is possible to instead have a rule where both must be within X points of each other.

-Advantages vs. skills:
What is a skill in one system might also be an advantage in another, or vice versa.

Advantages may be distinguished from skills for a number of reasons:
-skills being in multiple levels vs. advantages being purely you-have-it-or-you-don't.
-skills representing learned abilities while advantages are innate (WoD merits)
-different level of abuseability e.g. combat/noncombat (3E).
Or they might not be. HERO skills are a separate category but may as well be a sort of 'advantage' within context of that game, for instance (purchased the same way).
Mercenaries, Spies and Private Eyes has things that probably should be 'advantages' including "Title", "Secret ID/Double Agent" or psychic talents but that are just part of the skill list; the latter while purchased as a skill also require extra checks (Int roll) to see how useful it is. Rolling advantages into the skill list this way does mean it doesn't cover 'disadvantages' (all skills being positive), and for MSPE means INT (which gives # skill points) is paying for a character's title or secret ID.
Some abilities can be framed fairly ambiguously as either skills or advantages - contortionist vs. double jointed, seduction skill vs. 'attractive'.

Characters may also get fewer advantages than they do skills, which means that making something an 'advantage' makes it rarer. A good example of this would be Advanced Fighting Fantasy: in the first edition of this, the special skills 'Dark Seeing' and 'Strength' (giving +1 damage rolls) were both skills, but so powerful that nearly every character would have them, bought from a number of Special Skills equal to starting SKILL (of 7-12). The second edition of AFF added rules for 'talents' (=basically feats) and made the equivalents of these - 'Strongarm' in the case of the Strength damage bonus- both talents. 2E AFF characters have only one talent initially, and a larger list of talents to choose from, make these capabilities much rarer.
Advantages handling skills badly; Gurps 3E has a 'literate' and then also a 'semi-literate' advantage; I gather it halves reading speed.
*Quite often a game with detailed skill ranks will have some learned abilities that are yes/no, e.g. languages in many games, or 3.x D&D Feats.
WHFR has 'skills' including a number of things that would be specific abilities in other systems. MasterBook lets a character take 'unable to speak campaign area language' as a specific disadvantage, while another game would define that with (absence of) the relevant skill, in a more consistent fashion but greatly under-rating the impact of lacking the skill choice. Summerland uses 'tags' that can include either advantage/disadvantage or skill type abilities, which then modify the attribute checks used to resolve all game tasks.
*advantages vs. classes: Savage Worlds has 'Professional Edges' which are part of the advantage system but (on top of the skill system) replace classes.
*advantages vs. class features: what's possible here depends on how valuable a 'class feature' is, vs. an advantage (e.g. feat).
The relative value determines in which direction this can go. For instance, a 3E cleric has "domains", which include bonus feats, making a domain largely as or more valuable than a feat. That mostly disallows there being a feat called "Extra Domain" (unless its argued that the specificness of the feats available with the other domain powers offsets their actual value).

The presence of disadvantages may act as a factor driving consideration of how skills or attributes are scaled. For instance, skill may default to 0% in one system while another has a higher default, with complete inability still possible for characters but modelled as a Disadvantage (e.g. 'All Thumbs' in Savage Worlds for the truly 'Savage' primitive, unable to use mechanical devices).
Some 'advantages' can be duplicated by built-in systems. Some games for instance have an advantage raising a characters' purchase limit for an Attribute; in a sense this is not much different from just costing more points for a very high stat, apart from any psychological effect to the player (they know their character is considered superhuman).  

Disadvantages frequently have some sort of cap on how many points can be taken (e.g. GURPS has a variable 'disadvantage limit' set by the GM, while Savage Worlds automatically sets a limit of 1 major/2 minor). A disadvantage limit reduces the necessity of having +1 advantage be worth 1 disadvantage, although making disadvantages not-really-serious does handicap characters who opt not to take any.
Disadvantage limits may need to be higher if fairly ordinary build features require disadvantages to select e.g. is disadvantages are needed to 'buy down' attributes or skills, vs. what would normally be allowed with free distribution of points.
GM oversight may also be required; a game may assume that either the GM will monitor abusive combinations in character generation, or that they will adjust game play so that Disadvantages will be engaged in-game no matter how bizarre they are (this last is incompatible with some playstyles, and may require more GM oversight of hindrances taken). Some games cost hindrances based on frequency of occurances (i.e. Fuzion on an ad hoc basis, or HERO -4E anyway - had a random roll for each disadvantage on 3d6 to see if they would occur in a session, with higher frequency meaning more points) - this is incompatible with some playstyles where the GM engineers situations to deliberately exploit character weaknesses (the higher bonus points for commonplace hindrances is irrelevant - you may as well have taken the uncommon weakness to Kryptonite for more points if the GM was going to work it in anyhow). Disadvantage lists sometimes include minor things such as 'quirks' which really have no major negative effect, but act to reward detailed character concepts with more bonus points.
Disadvantages may sometimes (like some advantages) have some sort of prerequisite to select; this can be counter-productive if the prerequisite's cost uses up too many of the points generated by the disadvantage.

Characters might be allowed to add advantages or disadvantages after character generation, or not. Most allow disadvantages such as missing hands etc. to be gained from play, with no gain in points; Unisystem allows characters to voluntarily add disadvantages with GM approval for points in addition to this ( e.g. 'gaining levels of deafness due to off-screen practice in heavy metal bands'). If some can be bought off or gained also depends. In some games e.g. Supernatural the game specifies that some advantages will require 'story' reasons, but with the assumption the GM will make adjustments to the story to facilitate this.
Some systems might also allow characters to replace a disadvantage that no longer works with a different one, likely with GM permission that this makes sense - the character who doubts the supernatural taking a vow to destroy demons after realizing they exist, a character who loses a ward instead having depression, the battle killing a superheroes' arch-nemesis being violent enough to get federal agencies after them. Probably this works more easily in a game where many or all disadvantages have the same cost (e.g. Savage Worlds with its minor [=1pt] or major [=2pt] disadds. Buying off a disadd is relatively easier if it generates fewer points compared to character advancement i.e. if a character could pay it off in a session or two (again SW) instead of six sessions (a serious GURPS disadd being maybe six sessions of CPs) although the GM might be forced to allow prespending of future CPs if removing it has to match in-world considerations.

Potentially a damage system could tie into disadvantages/advantages, with X damage giving a character a -X flaw (particularly for something physical like 'scarred' or 'one eye'). The closest equivalent in a system I know of to this, however, would be Drain in Champions, which reduces stats or powers with an effect measured in character points. (One quirk of this method being that presumably abilities bought with cost-reducing limiting factors would be drained more quickly).

General Note: From a player POV, advantage/disadvantage systems can be a useful way to 'buy out' of aspects of a game system with which they are not entirely comfortable. Something like 'charmed existence' can reduce 'botches' if a player doesn't like that, or advantages can eliminate wound penalties, perhaps increase base HPs, and in various other ways let a player play a character similar to a different system's PCs. Avoiding advantages/disadvantages completely might let a player deal with less complexity, if desired.

General Note II: what a system's advantages/disadvantages are built around is (as with any character ability) determined by the level of detail in other aspects of the rules, in areas which can include things from movement/miniatures to skill use/advancement to game world physics to personality traits. The more detailed the rules, the more advantages are possible/needed for an area and the more likely a character will require mucking around with advantages to build it properly, particularly if its unusual.

General Note III: advantages are generally more useful to provide 'fine-detail' customization of a character. I think if a game has a huge diversity of options, people are more likely to play more 'steoreotypical' characters, whereas if there are only a few they are more likely to want to adds some mods and customize them.

General Note IV: as can be seen in some examples above (Synnibarr, ROAR) advantages are sometimes bought with abstract points, and sometimes may have a number based of e.g. an attribute. Non-abstract purchasing indirectly limits what can be put in as a reasonable option - the advantages will be something someone who is higher in XYZ should have more of - if most advantages are skill-like and purchased with Int points then a 'toughness' option there looks weird for instance.

See post 120 for some more notes on point systems in general.

The subtler the sorts of effects that Advantages/Disadvantages cover, in general the more of them an individual character will have to have, in order to be well-described within the game mechanics. Of course, this is also affected by how much heavy lifting is being done by other aspects of the character, like attributes and skills.
In simpler systems or those without a lot of rules, advantages/disadvantages may be just rated descriptively, with GM fiat applied to determine penalties or effects. For instance, a 'colour blind' disadvantage might just note that a character has trouble seeing colour; it would then be up to the GM to determine that the character has trouble on a particular check, for example cutting the correct wire on a bomb defusing attempt, or that a given target stealthing past them will be harder to spot because their colour contrast vs. the background is less significant.

A list of GURPS advantages [4E] is here as part of the GURPS Index - and maybe useful as somewhere to look in designing a list of advantages.

Games with advantages/disadvantages probably usually list 'advantages' first, then 'disadvantages'. Savage Worlds lists Disadvantages first - probably because to take any Edges you need the points from the disadvantages to do so (apart from human bonus edge) unlike other games where by default there's a large pool of points to begin with. It would be interesting to see what effect the order has psychologically on new players - disadvantage first may be a disincentive to advantage buying, since players won't be likely to not see anything they want and haven't yet seen what goodies can be bought with those points, although there would be some flipping to-and-fro.

Advantages/disadvantages might modify attributes, but its not necessarily a good idea since such an option is 'redundant' (maybe generating cost breaks which are undesirable). For instance, Gamma World had a "mentally defenseless" mutation that could be rolled and gives a character a Mental Strength of 3 - a character could just roll a 3 for Mental Strength normally, so it doesn't add much to the game.

[As sort of noted under 'controlling bonus and penalty accumulation'] Tag-based definitions like 'advantages' sometimes give better control of results than abstract numbers. For example, if a monster has the "hideous" flaw, then its hideous. On the other hand, if a monster rolls a d6 for Physical Beauty (vs. 3d6 for humans) then it'll be usually hideous, but a player might be able to stack some random bonuses to make it normal looking, like how Palladium supplement Mystic China has a martial art giving a +1 to PB for good poise and acting skill, etc. - probably this shouldn't work if you're a slime-dripping alien worm, but this and if you could find a few similar things could fix a low result. The problem being that a thing intended to give about a +10% to PB, instead give a bigger proportional shift, and enough could fix a low result.

Design Note: Like other character features, advantages/disadvantages can be written as more- or less-specific in order to represent a wider range of possible characters. (e.g. Pisces d20's Negative Feat Awkward Body Shape [the main game effect being preventing a character from wearing some items e.g. armour] - "Maybe you're portly. Maybe you're shaped like a letter of the alphabet. Who knows.". Another game might have a specific obesity disadvantage which also has Endurance modifiers, etc.

Backgrounds
Backgrounds are another facet of the character e.g. in games such as Storyteller, or FantasyCraft also has a detailed system (for renown, lifestyle, contacts, etc). In games such as Storyteller backgrounds are separate to attributes, merits/flaws and advantages/disadvantages, although in other systems they may just be a part of the advantage/disadvantage system.  Backgrounds such as resources, mentors, followers, items, reputations, and so on in a separate 'pool' of points is I think a worthwhile concept, if its tracked mechanically, since these points are potentially gained/lost more easily - through campaign events - than are stat points, and perhaps skill points. Compare this to say, GURPS, where reputation changes and so on must be 'bought' with character points. White wolf didn't however handle negative backgrounds (being hunted, having a nemesis, etc.) as part of the background system, they were simply normal 'flaws' (disadvantages).

Apart from these many games simply do not have a background system - the player can write up a background that makes them nobility or wealthy, or even grants them a magical item or similar, and its fine as long as the GM approves it. The GM might also add 'complications' to balance things of their own accord i.e. an ancestral magic item comes with an evil group who want to steal it. The overall effect here can be the same as if points were used (Wanted hindrance used to buy a special magic item) but with less accounting involved. Even in point systems, things such as nemeses and being hunted are perhaps questionable balance-wise - the game is likely going to have some sort of nemesis, so it makes little difference to the character if this is due to their Flaw or just assigned to them/the group by the GM. Sometimes a particular background feature is represented by a system without an overarching system for this e.g. a game might roll/let the player choose the number of Contacts that a character knows initially, or randomly determine Social Status (AD&D); HackMaster has a detailed Honor system (which also modifies attribute improvement rolls).
 
Some systems also have non-point-based backgrounds which are essentially random roll e.g. Cyberpunk had a series of LifePath tables (IIRC, fairly good). Paul (Jennelle) Jacquays authored a series of 'Central Casting' books which had guidelines for conversion to various systems, which were quite interesting. These could strongly affect the character including random boosts, impairments or extra skills (one of our games we got demigod once...), or results could be odd or contradictory.

13th Age lets characters choose 'one unique thing' (being a special snowflake is mandatory). This can't be combat-related or offer a 'hard mechanical bonus' (without costing other character resources at DM option, such as trading in class or race features) and could include odd physical features (clockwork hearts, opal eyes, human-eared elves, cutting off own arm to show how tough they are), odd racial affiliations (only halfling knight, human cleric of the dwarf gods), or an unusual relation to one of the settings' major NPCs (fated to kill the lich king, probably in a future age).

Apocalypse World /Dungeon World has players choose 'bonds' which define how the PC is connected to other PCs (a formal system for something that probably often happens in other games unprompted).

Both advantage lists and background feature lists for a given game will be designed to support the style of play for the game in question - preventing creation of characters that are unsuitable to the expected genre and playstyle, and costing options in keeping with their likely usefulness. If a campaign's style of play deviates from that expected, then, options will be miscosted, inappropriate, or the lists will fail to cover options (as unsuitable) which actually would be helpful.

(Note that the section above was written before 5E D&D, was so my use of 'background' is slightly different to what 5E means by a 'background'; I might classify that as being simple 'lifepath' system, cf. next post). 5E backgrounds are noteworthy however: it provides a very general sort of background pre-adventuring, as well as including some skill picks, a special ability and a set of possible personality traits to choose or roll. Mechanical dividing up skills this way would perhaps be a good way to reduce the amount of 'stuff' gained by multiclassing -  though they also decided to do that by not giving out all abilities when multiclassing.
See here for therpgsite's big list of homegrown 5E backgrounds.  While the background is a package, it is possible to build your own, and its also possible to 'swap out' specific features - for instance some adventures provide an alternative special power that can replace the standard special ability, that also being convenient if you are building a do-it-yourself background.)

Character Backgrounds in a general sense - rules interactions
Here I may talk a bit about general background design. To have a particular sort of thing written into the character's backstory, is affected by other rules. Fairly often even if there are no mechanic options, a GM can kludge something, but mechanics can streamline a process or even more so, can accidentally prevent an idea from working. Below is a list of a few fairly typical high fantasy ideas along with what makes them work. For example a character that's "secretly a necromancer" might be perfectly OK as a Savage Worlds character (one I've actually seen, since the magic skill costs only a few points and an Edge) but isn't so feasible in 4E, where a character uses magic nearly nonstop in combat.

Character background or trait      Mechanics/rules probably required (built in or GM fiat)

Adopted by other race------------   Swap native language for other races' language
Monster e.g. demon ancestry----- Monster powers e.g. demonic abilities
Conscripted to army-----------------Weapon proficiency (classes like 'mage' may cause problems)
Wanted / Hunted---------------------Disadvantage or GM fiat
Hunted by BBEG for owning object-----Hunted-type disadvantage, purchase magic object
Found/inherited a pet ---------------Animal companion rules
Scarred in accident-------------------Check Regenerate rules interactions (i.e. not worth free points)
Other physical defect-----------------Justified via stats (e.g. low Per - glasses), or specific defect
Noble title-------------------------------Advantage or ad-hoc GM permission.
Secretly a magic-user-----------------Viable without using magic i.e. low-cost/low-effect, or point discount for usage limitation
Personality issue   -----------------------Disadvantage, or XP for roleplay bonus.
Cursed-----------------------------------Disadvantage or GM fiat
From other region----------------------Invalidates region-specific skills.
Illiterate----------------------------------Disadvantage or point cost to learn reading   
Unusual size-----------------------------Size rules, or modelled via stats e.g. high STR


Allegiances
Characters can also be signed up to various organizations as part of character generation.
This can be a purely roleplaying or background thing, likely to have an impact on the campaign;
Or it can have a game-mechanical effect, sometimes directly and sometimes indirectly (that is, a particular background will set some limits on what can reasonably be bought, or at least raise questions or require additional explanations for some choices).
Some games where there are game mechanic effects have allegiance occur as a part of character class (e.g. in Mage the Ascension, your starting Mage group determines your style of magic and free Sphere dot); in others the choice of allegiance is a separate decision with minor perks e.g. in Gamma World, a character of most classes can choose from various Cryptic Alliances; in 2nd Ed. AD&D Planescape, planar characters could choose a 'faction', instead of choosing a 'kit' (class specific class add-on) like 'prime' characters.


recent edits: added 'allegiances' section(*); extra note on DIY'ing 5E backgrounds; reorganized adv. vs. skill/attribute (*)

Class and/or Skills
The known methods of determining PC capability include class-based (1), skill-based (2), lifepaths (3) and archetype (4).
 
1) Class based; a character’s class determines their capabilities and give characters “niche protection”.  The main benefit is to encourage diversity in characters, and it tends to encourage teams of characters (most often class systems work optimally with about 4 players). 2E D&D did often encourage the GM to run single-class campaigns (e.g. where everyone is a fighter), and potentially even where characters all have the same 'kit' (all barbarians, all gladiators, etc); this in retrospect seems peculiar in that it made characters very similar with only slight differences in ability scores and/or proficiencies and/or race. Advantages cited for that in Creative Campaigning is that it avoided 'spotlight time' problems where say the thief 'could do little in wilderness games', and that it was 'more challenging' (players needing to find innovative ways around areas of ineptitude); I'd note it also somewhat 'balances' a party e.g. everyone has similar ACs/HPs (something achieved in later editions like 4e/5e by fine-tuning numbers between classes), as well as limiting the effect of 'optional rule' rules variation which often had class-specific effects.

In OD&D, Dragon Warriors or other skill-less systems, each class can be thought of as a single very broad “skill”; skill systems are frequently added in later editions with some classes getting free skills, altered costs of skills, or banned skills. In Talislanta, each “class” is defined almost wholly by which skills it gets either as primary skills (at the character’s full level) or secondary skills (at half level) (though classes are also racial, and so have different racial abilities). Class can be very simple (e.g. a Tunnels and Trolls character in the adventure Dargon's Dungeon can have their class magically replaced with a different one and -aside from spells- be ready to go virtually immediately) or can support extensive arrays of dependent numbers and specific ability selection trees [e.g. FantasyCraft]. Hollow Earth Expedition has 'archetypes' like Academic or Big Game Hunter selected from a list as the first step in chargen, but with no game mechanics, even recommended skills - they exist purely for role-playing. Savage World 'Professional Edges' and FATE 'defining Aspects' also sort of give some class-like definition. A set of classes might be designed with standardized features (...they all get one bonus skill, a +1 to a particular stat, etc.) for balance or whatever reason, though taken too far this might cut into differentiation between the classes (it could be argued that 5E D&D does that it ways, with class features designed to e.g. provide unarmoured AC, or healing via second wind vs. spell, etc. reducing differentiation). Classes might have a sort of taxonomy or tree structure where several classes form groups; one interesting thing about this is that what goes where varies from system to system e.g. Techno is a Tunnels and Trolls 'warrior' subclass (Sorceror's Apprentice) as warriors can't use magic, while in Gamma World the equivalent (Examiner) is a main class, alongside Enforcer as pure warrior. Bard might be a rogue sub-variety or basically a singing wizard, or rogue itself might be considered a warrior subclass (e.g. 4E D&D has rogues as 'Strikers' using the 'Martial power source').
Classes might have various numbers of 'special abilities': a 1st-level character might get only one ability (Star Wars SAGA), 3 or 4 'powers' [4E D&D], or a large array of special abilities (or spells)(Rifts). More abilities relates to more competent PCs, but also make multiclassing more powerful and hence problematic.
 
Classes may have minimum stats to select (sometimes including requirements in e.g. social class, such as in 1st ed. AD&D with Unearthed Arcana, or alignment - this last possibly needing to be maintained to continue in the class), or be limited to certain races.
A racial requirement might be waivable in some cases - for example, if cultural and they're raised by a different race, or have a specific advantage/feat.
More rarely, classes can have attribute maximums e.g. a 1E AD&D barbarian can't have a Wisdom of 17-18 (the 2E barbarian class from Complete Barbarian has no max., but instead has a penalty to Int and Cha checks outside their homeland), while a Deluxe T&T character who rolls triples is a 'specialist' and so can't be a 'citizen' (the peasant class). The need to roll triples, rather than any particular scores, also means specialists can't be created with point-buy.

Attribute minimums can work to randomize whether a character can belong to a particular class in a random-roll system. Some ability scores may exist almost solely as a way of determining if a character can belong to a class (e.g. Wis in AD&D), being useless to other classes; this is probably a problem once point-buy or rearranging of stats is possible since characters of other classes can 'dump' these scores.  A low score might fully prohibit becoming the class or just make it a subpar choice.
Other random rolls can also be used to determine 'class' (usually fully yes/no rather than merely handicapping):
- Warhammer does this to an extent (although this is less of a problem in WHFR since a character is expected to change profession several times).
- Synnibarr uses random rolls to determine what classes a character can select from initially, but gives a character 3 rolls (rerolling results for which the character can't meet class minimums); it is also interesting in that it has characters becoming multiclassed when they become immortal at 51st level (even though it uses race-as-class), with the new class progressing instead until level 100; rare 'immortal born' characters (with 20 in every attribute) can start with two classes initially. Characters can pick their 2nd class at 51st level freely.
-Dungeon Crawl Classics generates 'profession' randomly although this really mostly just determined 'skills' and limits access to racial classes; the initial random system paired with random dice roll and anomalies such as Str 3 Blacksmiths, although fixes were proposed such as a set of tables, rolling based off highest score or (a compromise) weighting a roll toward the highest attribute(s) and then branching out to a separate table for each score - see http://www.goodman-games.com/forums/viewtopic.php?f=72&t=34279&start=25.
-Spawn of Fashan randomly rolls a 'parents' choice occupation' and then has a player choose an occupation, before stats are rolled; i.e. if stat rolls would make a character terrible at the chosen occupation, they can instead choose the random one (only). The exception is that if 'misfit' is rolled, that's automatically the class.
-in other systems, random rolls may modify class choice in exceptional cases, such as a character in 2E D&D Complete Celt rolling a 'gift' that let them be a human multiclassed wizard (humans couldn't normally multiclass, only dual-class); a low Social Status in Unearthed Arcana 1E D&D would prohibit being a cavalier (or force the character to start at "0-level", but not most other class choices.
-if race limits class choice, randomising race will then also randomize class choice (no known examples).
A class might have a feature with randomly-determined power, as well. One sort of example here is the Dilettante 'kit' in Dragon magazine for half-elf fighter/mage/thieves, which rolled 1d6 for how many 'secondary skills' were known (characters normally get 1 or very rarely 2). For me at least the point of the kit would be more skills, meaning the characters total resources (overall character design) can be wasted on a poor roll. That sort of thing - with major class features - can be an an issue in that since the main purpose of classes is to offer 'niche protection', they either fail to fill the niche or may be overshadowed by other characters on the basis of the roll. This sort of randomization isn't that different to having a # based off a stat (e.g. the kit would've worked if it gave Int/3 secondary skills) in that this provides variability, but known in advance. I suppose the counter-argument for this sort of randomness is that any extra variation could appear in a low-stats character and make them more viable.

Classes often have a single key attribute. Some systems deliberately weaken some classes by making them deliberately multiple-ability-dependent (MAD); many 2E D&D classes have higher requirements in stats which aren't 'prime requisites', while 3E D&D tries to weaken some spellcasters (favoured souls, archivists) by splitting spellcasting functions across multiple stats. In random-roll games this sort of thing makes MAD classes harder to roll up (rarer), while in point-buy it leads to characters that drop 'dump stats' significantly. Having a classes' abilities be heavily attribute-dependent e.g. the 2E D&D psionicist which uses ability checks for power activation, the 3E D&D marshal which gets auras adding +Cha mod to various things, and the the SenZar rogue - see next post) can be a problem as this front loads abilities rather than having them be level-based (at least its a problem unless all classes are designed similarly and multiclassing is restricted). Sometimes there may 'workarounds' for some ability score dependencies - a couple of 2e kits in 'Champions of the Mists' changed a characters 'prime requisite' (so letting characters get +10% xp more easily, if the class had more than one), or 3E has some feats or racial features that can change key abilities for attack rolls, skills, or class features. These are sometimes designed to make particular races (with a penalty) or multi-class combinations more workable.
One thing to note when considering classes' design is that the usefulness of a class is often determined very much by general rules of the game, rather than the rules of the specific class itself e.g. consider how loose skill rules helped the AD&D fighter work as a general adventurer without needing many skills, or how a Savage Worlds "rogue" is potentially very effective since its relatively easy to one-shot opponents. Conversely a class can be crippled if its shtick is heavily opposed by other game rules or factors (e.g. how 3E swashbucklers lack significant 'swashbuckling', due to tight system constraints on movement, and unusual combat options requiring feats to perform).
A few (older, especially) class-based systems are set up as such without providing enough diversity or niche protection to justify this - Rolemaster for instance distinguishes classes mainly by skill purchase cost; Palladium's Ninjas & Superspies has classes differentiated only by bionics budget, number of martial arts, and number of skill programs - classes resemble just different purchase arrays of those, and could be replaced with a single purchase system, if not for compatibility with the rest of Palladium.
More recently, Star Wars SAGA (the more recent of the d20 Star Wars) has classes with similar numbers between them and somewhat overlapping skill lists. The main differentiator are 'talents' specific to classes but the game uses 3E multiclassing anyway -choose a level of whatever class you want whenever you level up- so overall you basically are just picking whatever talent you want when you level up anyway. Classes with more skills have less weapon proficiencies (e.g. Soldier) but as both are purely yes/no (skill training = +5 to a skill) rather than 3E D&Ds varying # of skill points/level, this could also be duplicated simply by making a weapon proficiency cost a skill.

At the other extreme, classes in a complex game can be designed so that different classes interact with/get power from different game 'subsystems' e.g. FantayCraft (Courtier: currency systems, Keeper: skills, Soldier: feats). Heroes Unlimited's different power categories might be another example of this. This sort of structural difference - tying into various subsystems instead of one or a few - makes little difference ultimately, with how well this works depending a lot on the specifics of the implementation of the subsystems. One game might handle several things with the same subsystem that are distinct in another - a character might get the same game benefit from an attribute raise in System A, a skill in System B , or a feat chain in System C e.g. compare having to choose between more skills and having psionics, vs. psionics being a skill (classes can also use different 'rolls' or subsystems for checks to other characters - frex. some later classes in 2E like the psionicist rely on d20-under-stat ability checks moreso than other classes, which were designed mostly before there was an ability check system).
Classes may be differentiated by specific handicaps as well as special abilities (e.g. poor hit dice, weapon selection, etc.). T&T (e.g. Deluxe) gives a named negative feature to (most) classes (weapon inability, magic blindness or half powered magic). D20 glut era Pisces RPGs had a series of classes which each had one bonus feat and one balancing 'negative feat'. 3E D&D barbarians had illiteracy as a special feature, while Pathfinder Oracle's have significant debilities (e.g. blindness); 3E prestige classes in particular struggle with this a bit - characters are meant to want to aspire to join them so one eye [Eye of Gruumsh], a writ of outlawry [Outlaw of the Crimson Road] or sudden loss of all knowledge ranks [War Hulk] is a hard sell.  
The overall design of a class across its various aspects may generate significant differences between different classes (warriors being good at one thing, wizards at another, etc - as in 'old school' D&D/AD&D, or classes can be deliberately designed to be similar enough that they essentially only provide different "flavours", while all being equally able to contribute to combat, skills, etc. [4E D&D strongly, 5E D&D to some extent].

Multiclassing is tricky in class-based games. Sometimes multiclass characters fairly normal with up to several classes e.g. 3E D&D, while in other games they exist only rarely (AD&D dual-class) or under special circumstances (e.g. Rifts' retired juicers or T-men from Atlantis). Multiclass characters might be equivalent to a more powerful/higher level character, have ability requirements to qualify, or be normally against the rules except in very rare cases [e.g. NPCs]. Multiclassing can also be balanced by giving it to everybody i.e. Gamma World 7E inherently gives a character two 'origins' (something like racial classes); original Recon gives characters a 'primary Mos' and 'secondary Mos' so they're also sort of inherently multiclassed. (In this sort of setup you can design extra classes that let someone emulate being single-classed by just giving more benefits to the primary class, say if Expert/Artefact User is 'fighter with magic sword' or Expert/Paragon the just really strong fighter ).

I think generally you could say there are three models of multiclassing depending on overall structure: 1) you get one class and can pick up a couple of minor abilities as a sideline (4E multiclassing); 2) the character splits their capability between both classes fairly evenly [4E hybrid characters]; or 3) you get the full set of powers of both classes but with other drawbacks, like slower levelling [AD&D characters]. (3E is somewhere between 1) and 2) depending on how you split up the levels).

Multiclassing is sometimes done by splitting XP between two classes (AD&D, Palladium Fantasy), or splitting levels between classes (Japanese RPG engine 'SRS', translation here, mentioned in this thread, gives all characters 3 levels to split up initially). A more well known example, 3E D&D splits a variable character level between classes; some combinations ended up very weak in this setup, giving rise to patch options in Prestige Classes like Eldritch Knight, or feats that allowed specific classes' levels to stack for some abilities;3E also had an entirely different optional multiclassing system ('gestalt') in Unearthed Arcana which however powered up the game significantly and worked only if everyone played gestalt characters. A level-splitting system also has a 'feedback' effect on class design - it severely constrains the number of abilities that can be given out at 1st level, since front loading encourages characters to multiclass excessively (a problem with the 3.0 ranger).
Other systems deliberately added new classes to cover particular combinations, though this eventually leads to a lot of classes. Rolemaster (which let characters of any class choose any skills) sometimes just allowed characters who 'multi-classed' to alter development costs of skills to the average of both classes. Multiclass options are sometimes limited by race or ability requirements (in original Palladium Fantasy multiclassing is slightly tricky as a character needs to meet both sets of class prerequisites; AD&D dual-classing or 5E D&D multiclassing has extra-high stat requirements). Multiclassing may also have serious limitations due to game balance concerns; in some cases it may be perhaps deliberately not be included as an option (e.g. Rifts, where character abilities are heavily front-loaded compared to level-up benefits; multiclassing is sometimes necessary e.g. for slave-borgs or retired juicers, but is handled on an ad hoc basis. Characters can also sometimes pick up individual archetype abilities like psionics, super powers, martial arts forms or cyberware; each case has its own custom rules).
If multiclassing is relatively common or easy, character concepts can be handled with the multiclass system which would otherwise require more complex & specialized individual classes to cover them. As in Synnibarr or Apocalypse World/Dungeon World, multiclassing (at least picking abilities from another classes' list) can also be used to avoid writing as many abilities for one class (e.g. in Synnibarr, as noted above, the character gets a second class at level 51 with 1st level abilities; this then has a 1-50 progression in the second class to cover up to level 100 without having 'dead levels').

Sometimes multiclassing does not grant all benefits of a second class (e.g. in Fantasycraft, the 'core ability' of a class is only gained if its the characters class at 1st level). 5E reduces number of skills gained by multiclassing by splitting them between 'background' and 'class' - a 1st level character has 1-3 skills +3 from their background, giving them 4-6 skills total but only another 1-3 if they multiclass again (since a second background isn't gained), as well as (in the final version) not granting the full set of save proficiencies for multiclassing.
Chris Perkins's "AD&D3E" has primes (derived from C&C, I think) - a character has two primes (three for humans), one of which corresponds to their class and gives +5 on relevant checks; multiclass characters will just have their focus areas (equivalent somewhat to class specialties) preset, rather than gaining any extra bonuses.
Very limited multiclassing may give only a single ability from the second class, as in 4E D&D power swap feats, Conan D20s 'Arcane Dabbler' feat, or in 2E Skills & Powers via some of the alternate class options like 'warrior-priest'. A GM may also let a player just 'swap' an ability from one class, for one from another (Skills and Powers did show that this could be a problem when a class ability didn't always kick in - for example, if a class feature requires a high ability score to use like 'fighter bonus to HPs', trading it out might be giving a 'freebie' bonus power). Sometimes a single bonus ability comes with handicaps relating to the original class e.g. Skills and Powers would let a fighter get a ranger's animal empathy, but they would then need to abide by all a rangers' ethos restrictions, or a character learning a single spell but not being able to wear armour with it (like a mage).
Games while are partly skill-based may let some classes branch out into other areas at a cost by buying skills at greater expense (Rolemaster; 3.x D&D to an extent). Rarely multiclassing might add a specific disadvantage to a specific combination: in oWod Storyteller, Vampire and Werewolf types - while not 'classes' exactly - could be combined to create an 'Abomination', but this required a Gnosis roll botch to avoid death, largely limiting it to NPCs, and was later noted to also cause a deep depression limiting dice pools, 'hirano'(sp?).
Sometimes classes are only available as a multi-class option e.g. 3.x Prestige Classes, the 1st edition AD&D Bard, AD&D OA Ninja, or Palladium Fantasy's Alchemist class (NPC only, requires prior wizard and diabolist [rune guy] levels). Such classes may be designed as add-ons and not have the full range of abilities normally given to a class, e.g. AD&D bards don't increase their 'to-hit' values.
4E D&D has "paragon paths", "epic destinies" and "themes" which are expansions of the class system: every character may choose a 'paragon path' which gives extra powers at a appropriate level. Likewise a character can choose a 'theme' (e.g. Athasian Minstrel, Dune Trader, or Gladiator) and gets one theme-related power, plus at higher levels can select further 'theme' powers in place of normal class powers. Some themes, like classes, also have associated paragon paths.
A specific problem to avoid with multiclassing is to make sure MC characters’ aren’t better than single-class characters at their niche (or generally). (Watch out for dual-class fighter/illusionists who can keep fighter HPs and also get bonus HPs for phantom armor or whatever other spell)
Classes with more built-in flexibility can work better in multi-class combinations, e.g. the 5E D&D Warlock can be taken in a few different directions that would complement other classes, e.g. either combat or skills or magic. OTOH, adding flexibility to a class makes it need multiclassing less - adding Skills & Powers rules to a fighter/cleric, the cleric gained most of the fighters abilities (e.g. two weapon use, fighter Str/Con bonuses, using non-blunt weapons) making fighter/cleric less useful.
A couple of games have multi-classing as a power of characters with 'multiple personalities'. Spawn of Fashan treats each additional personality as having a separate (rolled) class, and gives each d4 'trigger phobias' which if a save is failed moves the character to a new personality. 3.0 D&D has a 'Thrall of Demogorgon' prestige class, giving a character multiple personalities that lets them multiclass without a favoured class penalty [arguably this would've been better as a Feat, since its giving a character a class than then lets them take more classes...when total levels to split are still limited].

Supers games sometimes use class systems for powers too with classes like Alien, Cyborg, Mutant, Psi..  - this is seen in Heroes Unlimited, Superbabes and a couple of D20 variants. One thing to note with these is that they can make 'secret origin' stories like the Savage Dragon turning out to be of alien origin rather than a mutant more difficult to pull off in-game, as well as making 'multiclassing' necessary for characters like Wolverine (i.e. mutant w/ bionic skeleton). The 'Invulnerable' RPG (point-based rather than class-based) specifically adds rules for 'mysterious origin' characters, possible since most origins are bundles of powers) with the note that a character whose origin is revealed may 'tweak' their stats. Aliens in HU are sort of inherently multiclassed, with a mix of alien-specific special features based off physiology + limited additional powers from the super-power, bionics, magic-user or other categories (one category per character only). Gamma World 7E is interesting in that characters inherently always have two origin types that they combine, making them fairly weird and inherently multi-classed.

Class systems are somewhat inflexible - a player might actually want just something simple like more skill points, more attribute points, more mutations or super powers or Edges/advantages which in a point-based system could just be purchased directly, whereas in a class system have to be purchased indirectly, through a 'class' created to allow that. From this standpoint, some 'classes' in a class-based game list will possibly be just kludges to get around the limitations of a class system (e.g. perhaps 'Experts' in 3E D&D). There is also often a 'class' created for characters with no particular training (i.e. 3E commoner, Rolemaster 'No Profession'; earlier D&Ds instead normally used Fighter as the default, and/or assumed inept characters were 'Zero level'. Games where class rules apply specifically to PCs only, such as 4E, don't have to include an 'unskilled' class).
Classes inflexibility also tends to make them more genre- or setting-dependent; most universal or multigenre games are skills-based rather than class-based (the exception being Palladium, which instead just writes a new set of classes for each game). Specific class abilities like 'turn undead', 'favoured enemy' or etc. can set world expectations, or force houseruling to replace features if a gameworld doesn't match the default.
Games can sometimes include rules for making up your own 'class' (2E AD&D); although even being able to do this, somewhat undermines the idea of their being classes. This resembles skill-based chargen a bit, except that building a DIY class tends to set advancement options permanently; skill-based characters tend to be more flexible. One site here includes a randomized DIY class generator for BECMI D&D. http://www.apolitical.info/webgame/class/becmi.php
Classes sometimes overlap 'attribute' as well as skills - D20 modern's "Fast Hero", "Smart Hero", etc. or Tunnels and Trolls has 'specialists' who are super-good at particular attributes. This can also be a multiclass option so someone could be a fighter/superstrong say (even 3E D&D in DMG II has a  prodigy special ability that isn't a class but replaces levels of one by having a level adjustment, and gives a +4 on checks). This can be a fairly cumbersome and fiddly way to let players be exceptional at stats - it may be better to design attributes to be purchasable at higher levels and innately provide whatever exceptional abilities you want.
Barring multiclassing class selection is either/or, so classes are usually balanced against each other (also barring particularly strong classes having attribute or race restrictions). Making something a class - things like 'merchant' or 'noble' for example - can therefore involve a lot of work to build extensive somewhat wacky abilities to balance them against pure warriors or wizards. That does make these work as 'pure' options, but its also possible a player would be happy having such abilities made into an advantage, skill, or other 'add-on' option. 2E AD&D for example has a number of merchant 'kits' (so a merchant will also have a class and so other abilities) while AD&D (Dragon #136) has a merchant class for AD&D which is available as a dual-class option regardless of race. FantasyCraft Courtiers or Star Wars d20 Nobles exemplify the other approach, where a full class is constructed for these. Providing 'niche protection' for classes designed around non-combat use can also make it harder for the combat-based characters to interact outside the combat minigame, leading toward the Shadowrun problem where game play is split between the Face, Street Samurai and Decker dominating different scenes.
Class variations sometimes split into 2 classes over time: the Rifts Borg includes 'partial conversion' and 'full conversion' options (much more powerful); Ultimate Edition instead makes the Borg class exclusively full-conversion with another class, Headhunter being an example of partial conversion.

Class abilities can form different amounts of the 'pie' of total character power in different games. Its interesting to compare 3E D&D and 5E D&D, in that limiting feats and making them optional in 5E tends to mean most of a characters' power is built into the main class level progression in 5E, whereas in 3E customization varied a character's power more depending on what feats etc. they chose. (5E characters vary much in skills since those are 1/2 based on background, but are more standardized by class in terms of damage output and strategies by which their damage might be generated)

Some games tend to not want much 'niche protection' in which case skills are a better option. An obscure example would be 'Monsters and Slayers' (reviewed in Dragon magazine) where NPCs might be just thieves or fighters or mages but player characters are 'Slayers' all trained as fighters, wizards and assassins - each has its own set of skills or spells and this is actually a skill-based game; it generates characters sort of reminiscent of (1E) Advanced Fighting Fantasy, where characters chose various special skills up to their Skill and always always including some thief skills, combat, and probably magic.  

2) Skill selection: characters purchase skills (a fixed number of skills or using skill points).
Simplest examples of this are almost similar to class systems e.g. Barbarians of Lemuria gives a character 4 'careers', rated at +0 to +4 (adding to 2d6 along with an attribute bonus, likewise 0-4). What tasks a career helps on is largely a matter of GM fiat ('hurtling toward a giant pit, Krongar makes an agility task..luckily the GM is in a good mood and decides his natural Barbarian instincts come into play, allowing the character to add Krongar's barbarian rank of 2 to the die roll').

Some systems may sometimes use "skills" for non-characters - e.g. Fate Core [the 4th edition of FATE]. See discussion of the Fate Core "fractal" here: the short version is that they define a skill as representing the ability of a "story element" to influence a scene, with story elements including potentially characters, locations, Aspects, or whatever).

3) Lifepath: characters roll a character background which determines their skills. GDW systems such as Traveller are the classic example of this; a modified version of this is Warhammer Fantasy where in play, characters continue to add new “careers” (despite characters basically being adventurers as an occupation i.e. class acquisition is pushed off to a slightly metagame level). Some terms of service may have prerequisites e.g. Starcluster 3E has stat or skill minimums, though with a 'waiver %' chance for a character to scrape in regardless (someone thinks they have potential).
(Edit 3/12/2014: see later in thread for discussion of non-random lifepath systems in e.g. Volant, where templates are layered up without random rolls: see page 22

3.x D&D can be thought of as layering a 'lifepath' system over the basic class system through its multiclassing rules.
Palladium in places combines random-roll skills with classes - mutant animals in TMNT roll up various skills, while 'Survivors' in Dead Reign (fairly uniquely among the classes there) roll to determine their prior occupation/skills. Survivors also roll d4 to determine their level in occupational skills, due to prior experience, unusual since Palladium OCCs instead get a flat percentage bonus.
Mutant Epoch has a single background roll rather than a 'path'; it has a row with different %s for each species, but ignores direct attribute effects (you're as likely to be a courtesan with Appearance 1 as 100). Customizeable species get multiple entries (you roll up 'bioreplica military' or 'bioreplica pleasure', or likewise military or 'comfort' clones, and each then get separate profession tables) rather than races being more customizeable or giving a profession-specific bonus.
A particularly over-the-top Lifepath system was found for HOL in its lone supplement, BUTTery HOLsomeness. This starts a character off with a 'totem' giving base stats and a roll on the Homeworld table, then characters roll on the "Chart Chart" to see which chart they end up on, including options such as Crime, Family, Generally Bad Choices, Specifically Bad Choices, Cornholed by God, Sidekicks, Mutations, Night School or Pressed Ham. Each chart entry gives an effect then directs to another different chart; generally good charts cost a point from a pool of starting pool of 'kudos'. A character can return to the chart chart by spending 2 points (a really good idea should you be about to go to the "Cornholed by God" table), or more to go to a specific table. Results give bonus skills/stats until a character falls below the last few points and is mustered out (converting any final points into stats or skills).

4) Archetype: as skill selection, but a pregenerated character with an array of skills is provided, ready to play (TORG, Shadowrun). Archetypes usually place no limits on how characters may progress after character generation.
Systems with extensive skill rules sometimes also have 'packages' of skills (perhaps a sort of 'sub-archetype). Rolemaster for example does this, with each packages having a different costs to buy for each class (because each skill has a different class-specific cost).
 
Interaction between Skill System & Class System:
A basic 'class' system functions as a set of extremely broad 'skills' which largely avoid the need for a skill system - basic task resolution can use saving throws and attack matrices, or more rarely player skill.
0D&D had no skill system; Tunnels & Trolls had originally no skill system per se, but still occasionally had skills as unique, one-off abilities - in its solo adventures a character meeting the Shoggoth can make an IQ and/or Luck attribute saving roll to be skilled in playing the piccolo (useful against shoggoths), or can be taught improved bow skills at the Archer's Guild in Gull.
AD&D through to 2nd Edition developed add-on systems for minor skills (e.g. Secondary Skills, Non-Weapon Proficiencies) which made it possible to run the game either with skills or without -  potentially even with a mix of skilled/unskilled characters, since a character not using NWPs might have access to abilities that would normally cost a proficiency slot at GM discretion (reading, swimming). On/off skill systems are also a topic of 5E D&D discussion, with about three possible systems that may be of interest discussed here:
http://forum.rpg.net/showthread.php?687366-5e-D-amp-D-Next-Q-amp-A
The final 5E also has optional rules for running it skill-less (DMG) - giving characters 'proficiency bonus' with one stat check based on class + one for 'background' (=actually free choice). Needs GM tinkering to allow for racial bonus 'skills' like elf perception. Similarly but less extremely, 3E D&D had optional rules that eliminated skill ranks and just had yes/no skill proficiencies (Unearthed Arcana).

Other class-based systems such as Rolemaster and Palladium used skills to define the classes, making them integral to the game - for Palladium magic remained unique to certain classes but a majority of class features including stealth (Prowl), hand-to-hand combat abilities, and weapon proficiencies were built as "skills", with different skills distinguishing e.g. the Mercenary from the Soldier.

Rolemaster took this further again with hit points ("Body Development"), magic use (spell lists) etc. all described as skills, and with characters having some ability to pick up skills from other classes.
In Rolemaster the main difference between classes was that development costs were different, making it more expensive to pick up unusual skills for a given class. RM classes are fairly similar, with specific class features modelled via particular skills or "spell list" options. Spell lists were however particularly expensive (e.g. RMC semi-spell-users such as Rangers paying 4x as much for a spell list) making (IIRC) wizard/fighters a better option than fighter/wizards (so to speak).

2nd edition D&D Skills & Powers has a system of 'character points' which actually gives a final result fairly similar to Palladium Fantasy's skill system. S&P has menus of class abilities for each of the main classes, with point costs for each - equivalent to picking skills within each class, although with finer detail in cost based on utility, and the same ability having different costs between different classes. S&P also 'prepurchased' future skills with the initial points at first level i.e. at first level a character might buy followers for 9th level (with a point discount for the delay), or paladin spells at 4th level, instead of getting extra points/skills from levelling and using them to buy the skill at level 9 or level 4.

3E D&D had skills covering primarily 'adventuring' skills, but with magic and fighting ability being outside the skill system. Classes provide large numbers of unique benefits at various levels, while skills are either 'class' or 'cross-cross' - classes vary somewhat in skill due to expected ability scores, and rarely due to class features giving extra bonuses, but largely a class is good at a skill or not (compare to Palladium where classes could get between +0% and +40% skill bonuses, or "Archetype" systems where a character might be built with some or a lot of a particular skill).

Notes:
*SenZar gives out different skills to different classes, but skills use only an attribute check and so never improve from levelling up. Consequently skill-based classes such as Rogue are traps, particularly as part of a multiclass combination - exacerbated by XP being split in half by multiclassing when skills are learnable just by training in downtime, so you can pay 50% of your XP forever for skills that someone else got by taking a few months off and spending some gold. The same basic problem of scaling is found with AD&D 2E bards, who regardless of level perform with musical instruments no better (as far as raw skill check goes) than a proficient non-bard with the same Charisma; and with a number of the NPC classes in 'Sages & Specialists' like Scribe or Blacksmith which for the most part are redundant to a character with the right non-weapon proficiency. This sort of setup can be made to work, however, if attribute increases are built into a class progression: for instance, Warhammer 1E/2E has a specific 'advancement scheme' for each profession.
*Earthdawn (1E) assumes all characters are 'Adepts' with magical powers. Characters can choose talents from other classes, but these are received as the non-magical version of the talent ("Skills").
*Savage Worlds has 'Professional Edges' that look a bit like character classes. These just add a bonus to skill checks (skills sold separately). I quite like this approach as being primarily free-choice but with a nod toward classes.
*A few games distinguish between classes, which usually provide level-scaling benefits, and "kits" or "specialties" which are one-off packages relating to character concepts. 2nd Edition D&D calls these "kits" and has separate lists for each class (and some multiclass combinations), while FantasyCraft has a single list of "specialties" which serves a similar role. D&D 5E has very broad mega-feats which might also sometimes be used to define subclasses.

Systems sometimes have problems modelling particular character concepts because these concepts rely on 'skills' which are outside what the system normally handles. For example, a "Tanis Half-Elven" character in 4E D&D could have a 'warlord' type class, whereas in 1E, tactical skills rely on using the player's wits, while motivating other PCs would likewise be roleplaying (or just use of Charisma). The thief class in 0D&D has some issues with its abilities duplicating things other characters could probably do to some extent anyway, such as finding traps by describing how this is done or sneaking; likewise diplomatic classes aren't overly useful in systems without diplomatic skills. The Demon Hunter in Mystic China is another interesting case since the class is meant to use trickery and deception to overcome demons, though there are no skills for this in Palladium; the class gets extensive roleplaying notes on possible tricks, plus some interesting items and a few other skills which are left there so a clever player *might* be able to leverage some advantages over demons as written, including super-drinking, play dead, and arm-wrestling skills, and a minimum Int of 10 so a player being cunning isn't role-playing too badly. Another interesting case is the Riddlemaster (Bard kit) for 2nd Ed. AD&D; this supplements player skill in riddle-solving by allowing two 'tries' -the first unsuccessful guess by the player didn't happen and is 'retconned' out as far as the game world is concerned- or double the normal OOC time to solve a problem.

[Edit to add July 2018] Where class affects skill availability, there's a subtle difference between point-buy vs. binary skills. With point-buy (e.g. 3E D&D), a character will tend to slowly pick up various skills covered by their class, while with 'binary' skills they're defined as knowing a skill or not; if your 5E ranger didn't pick Knowledge (nature) initially they'll probably never know it. The binary skills approach works well for classes which include several varieties (the rogue with a focus on being a con artist, sneak, lockpick, or pick pocket) but can also create sometimes peculiar permanent knowledge gaps.

Other
Vivsavage's Tower of Adamant RPG in development has a system of 'roles' separate to classes - representing a literary role like protagonist, guide, supporting personage, (a la Sam Gangee in LOTR), or anti-hero - these give various benefits (often metagame, like a guide being able to narrate change to a terrain feature). Roles can change/be exchanged after adventures.

Edits: Note on overall design (*); primes(*), classes & different core mechanics in 2E (*) note on Star Wars SAGA as having a poorly-designed class system, extra multiclass note; 7/7/17 multiclass structures overview. 19/11/17 - parties of all one class

Number of skills known by characters
Number of skills/skill points a character receives may be determined by
*character point expenditure (GURPS)
*fixed # points (Storyteller)
*character age (Ringworld) - usually higher age meaning more skills. Age can also have a 'lifepath' effect e.g. Systems Failure for Palladium gives younger characters different skills due to growing up after the Apocalypse; FGU's Daredevils, set in the 1920s or 1930s, has rolls for age with characters of enlistable age at the start of the Great War (depending on campaign) having to fail Health rolls or be drafted.
*attributes such as Intelligence (and/or Education or Willpower). (e.g. 3E D&D). 3E D&D modifies the class-based skill points from each level by Int. More directly a stat can just set # of skills: the Buck Rogers Cliffhangers game (according to a Dragon review) rated each stat basically 1-4 which determines how many skills for that stat are learned. Mercenaries, Spies and Private Eyes gives a character 'skill points' equal to the Intelligence attribute, with more learned only when a character raises the stat with levelling up (though known skills can also increase level up by earning XP).
Int bonus is more likely to apply if 'skills' are largely intellectual; heavily skill-based games where things like e.g. hit points are a skill, are less likely to give skill bonuses for Int. "Education" may also be a random roll that isn't a stat (Palladium, Heroes Unlimited) and/or affect if you can buy a skill (Medical doctor in TMNT requires a Masters or better education).
"Class" can affect which stat determines skills e.g. Amazing Engine generally uses [Intuition+Learning]/10 but with Once and Future King having castes which either use this normally (Technologer), Psyche+Willpower (Acolyte), Charm+Position (Courtier) or Fitness+Reflexes (Knight), somewhat justified by access to different 'skill groups' in each case.
(# skills based off stats can be argued to be reasonable/realistic, with the drawback being that terrible stats mean a character also has terrible skills. This may be unfairer in a random-roll system, then, but OTOH having skills based off a stat in a point-buy game allows for character customization to give a player the number of skills they want. Its a potentially complex question that has to also consider what the game calls a 'skill' - weapon proficiencies, spells, body development, stat raises [HARP] - as well as if/how much attributes otherwise modify skill checks. Variable amounts of scaling are possible based on stat ranges and formulae and can make or break the idea. Note also that pregenerated archetypes become more complex if skills vary since any stat change will need skills added or dropped [Alternity], and racial attribute modifiers e.g. to Int become more important [3E D&D]. ).
INT or similar typically doesn't affect number of 'skill points' in point-based games where you're just spending from a pool e.g. GURPS, HERO. This would be because points are split between skills and stats and so points spent in Int would be counted twice as it were. These games actually have the reverse - more INT means less points left to spend on skills, although only subtly and for GURPS, stats affect skill rolls so much it doesn't much matter. GURPS also has a pretty broad definition of what a 'skill' is, including fighting skills and the like where being a brainiac isn't necessarily super helpful.

*# based on class/level (3.x, Rolemaster/MERP; the latter has adolescence skill ranks, as well as those for gaining levels).
or a combination of these.

*Random rolls can also be used directly in various ways i.e.
-random roll to determine skills known (Marvel Super Heroes/FASERIP). 5th Ed. Tunnels and Trolls determines languages randomly (1 roll per Int point above 12) ignoring duplicates; Common is 50% likely and telepathy 1% likely. Early Gamma World had no skill system, but some randomly rolled mutations were sort-of really skills, like "economic genius capability".
-random roll for # points (e.g. CHILL).
If both # and which skills are random, a high roll might end up wasted if a 'roll twice' result generates one or two crappy skills.
-RECON (1st ed.) also determines skill levels randomly - after picking a skill a character rolls d100 to determine their skill bonus (no attribute modifier). (IIRC this looks painful in Recon with checks also being d100 against the skill rating?; linear rolls for skill 'rank' would be more balanced if Rank 1 is the base and extra ranks each give limited bonus).
- Synnibarr has characters roll to see if they have the talent to Sing if they buy the skill, with skill points spent indicating a 50% chance of gaining the skill - otherwise the points are lost.
Overall a random chance of some characters having an unusual number of skills can be given if characters either roll to see # of skills known, roll to see which skills are known with some rolls giving a 'roll twice' etc. result, or roll to determine the rating of a skill with some rolls exploding (e.g. crossbreeding Deluxe T&T's random skill ratings with Maelstrom's roll-up system: you could roll d6 to determine skill bonus, with a '6' instead being a d6 roll for two skills).
-lifepath type systems can sometimes determine initial skills indirectly. The generic Task Force Games "Central Casting" books are an example here, where characters would roll various background events including for birth, childhood, etc: this could include skills ("learn head of household occupation at rank 2"; "apprenticed when a talent for a skill is discovered" (more ranks if an apprenticeship event makes the character's master legendary in his craft); conscripted learn a weapon skill at d2; etc., as well as items, injuries, or curses. The CC books tend to slightly power-up characters if bonus skills were allowed. One game system that seemed like it would dovetail nicely with the system there was Mercenaries, Spies and Private Eyes which encourages the GM vaguely to give out a few skill points as part of character background, if its particularly good; hence, the CC book could plug in just by replacing ad hoc GM-permitted bonus skills with the random roll results.
-cost modifiers might be generated by random-roll. 'Dark Fantasy' rolls handedness randomly; left-handed characters get attack bonuses vs. dextral characters, but also training cost of skills from right-handed trainers is increased by 10%.

(Player knowledge can also influence a character's abilities in some games (such as roleplaying/player social skills where no explicit skill mechanic exists for convincing NPCs of things, player tactical acumen in games with fairly tactical combat, or trapfinding in games that rely on player description of how the problem is solved). Player abilities are not readily testable with game mechanics, so a default to player skill usually happens outside the game system. Character background and actual skills selected (if these exist) could however influence how much knowledge a player can legitimately bring to bear without this being seen as hugely or invasively meta-gaming.)

Depending on system a starting character might have 1 skill for example in Deluxe T&T (for a 1st level character), AD&D 'secondary skills' [in 1E DMG/2E PHB], or DCC character 'profession', half a dozen broad skills (5E D&D), up to 15 or so skills (Savage Worlds - 15 points, so a varying number of skills depending on their rank), or 30+ individual skills (e.g. a Rifts Coalition Military Specialist, for skills they have; or a Call of Cthulhu character has a character sheet listing their rating for every skill in the game, but with many being at a base rating representing being untrained).
Often if a character only gets one skill, it is very broad and its impossible to learn more skills; generally it becomes easier the less meaningful individual skills become.

At a systemic level, how many skills an individual character has is partly driven by:
-length of the skill list and hence how specific skills are (cf. also 'skill lists' below).
-untrained default system: if a default is very low (or conversely e.g. in BRP, very high - i.e. attribute check) this acts as a driver of a large number of skills so that most skills are trained.
-whether there's a major 'jump' between trained/untrained i.e. 0 and 1 rank of a skill. If "competency" is at say about rank 5, a character will be given more points (and might instead have lots of low-level skills). This is something in part determined by 'granularity' of the dice mechanic. (this is something I noticed converting characters between lots-of-skills and few-skills games; one way to do it is to just select the few highest-rated skills).
-what aspects of the game system are covered by mechanics e.g. what things are instead covered by player skill.
-whether skill-type functions are covered off by 'class' (or even attributes); some class systems use skills to further define what classes can do exactly, in which case # of skills may skill be quite large (Palladium).
-if skills automatically scale up with level, then characters may get fewer extra ones as they level up (the automatic scaling is the primary method of advancement). It might be possible to add a patch skill that instead gives more low-level skills, i.e. instead of one skill automatically at +X someone could take a 'Skill Group' skill that gives X minor skills at +1.  Although, automatically scaling skills may also pair with especially big skill lists (see Skill Lists below).

Scale of Skill Numbers
Unlike attributes damage is not usually assigned against skills, and they are not usually 'rolled up' randomly (in part I suppose characters are usually assumed to have some control over their career choices, whereas they don't over how healthy or smart they innately are).  Consequently (as there's no need to design the range to buffer high random rolls) skills are normally scaled to function directly as a modifier to rolls, (along with any other applicable bonuses). (One example of skill 'damage' would be games where the core mechanic is resource-based e.g. Gumshoe, Dying Earth. Skill modifier may also be used to derive numbers like 'magic points' or 'body development' hits indirectly).
Sometimes a skill number may be used as a bonus to damage, bonus to 'soak', or other effect, (needs fairly granular skills to do directly).

Skills may be binary (you have it, or you don't;  limiting divergence in bonuses, and preventing players focussing all points on the most powerful skills; it can also be for simplicity e.g. if skills are also being costed based on how useful they are), or be buyable in multiple levels.
If binary, usually a skill either defaults to a flat attribute check (e.g. 2E D&D NWPs, Warhammer, Amazing Engine) or has a bonus that's a flat number (4E D&D's +5 for skill training), or may have a fixed class/level determined value (e.g. Palladium, though classes also get % bonuses). Languages may be treated as binary even if other skills aren't, with other skills applying for checks (e.g. Persuade).
If a raw attribute check represents 'trained' use, then untrained use might be impossible, a 1/2 attribute roll, default to player skill, or largely unspecified? (2E AD&D). World of Dungeons (simple Apocalypse-world variant) has a system where a 'skill' (really, class feature) adds no bonus to a raw stat check, but a character with the skill can't fail skill checks, instead receiving a worse 'succeed but with complications' result; already high rolls get no further bonus.

5E D&D is interesting in that skills are yes/no but with an automatically scaling bonus - a skill gives a bonus equal to the 'proficiency bonus' (hence tying into other rules); saves and hit bonus also grant bonus equal to proficiency, making them essentially equal to skills for most purposes (though a bonus that can be applied to 'any skill' couldn't be added to a 'save').
Minor kludges on basically 'binary' systems can be seen in 2nd Ed. AD&D (where an extra 'slot' gives a pitiful +1 to the check) and Magitech for Amazing Engine (which lets characters buy 'half skills' for roll-under-1/2 stat, instead of full). Sometimes a binary skill can be boosted with an advantage e.g. 4E D&D skills are trained (+5) or not, but characters can also take the 'Skill Focus' feat for an extra +3.
Level-based games with binary skills have trouble checking core class functions (i.e. they don't scale with level, or don't consistently use the same mechanic, or could have complex 'trees' of very specific abilities with prerequisites - I think Deathwatch is an example).
Binary skills also don't work with skill-based games, leaving them a choice that best suits attribute-driven games. Some exceptions here are 5E and Palladium - in both cases while the skill itself is purchased Yes/No, but it gets an automatically-scaling, level-based bonus.
Talislanta [3E, I think] gives characters a number of skills that automatically scale (a class skill may be 'primary' and increase +1/class level, or 'secondary' and +1/ 2 class levels), but can also buy 'rudimentary' skills with XP.
A primarily-binary skill system might let some characters effectively scale up a few skills with advantages or even powers that boost the score.

Skills are also sometimes set up on the same scale as attributes; for instance Savage Worlds rolls either stat or skill (which are the same d4 to d12 scale), or FUDGE where both use a ladder of ranking adjectives like Fair, Good, Superb, etc. In both cases the skill roll is unmodified by attribute. Storyteller is somewhat similar (1-5 dot scale for both skills and attributes - or almost, since skills can also be 0 dots) though it does apply stat modifiers since a skill check rolls [stat+skill].
Overall, this approach can open up some options for more efficient rules, like for fair stat vs. skill opposed rolls or even stat vs. power checks (Savage Worlds' powers e.g. from Arcane Background have skill ratings), for giving free default skills (e.g. free native language skill rank = Int score). Or, from Exalted: 'Linguistics rating counts as Appearance for written social attacks'.
This is largely incompatible with binary skills (apart from the 2E NWP method), as well as significant level-based skill scaling (as attributes are usually fairly fixed).

Other specific oddities related to scaling:
Modifiers: an exception to the rule that skill rating = skill modifier, in Rolemaster a character buys skill ranks and the rank then provides a modifier to checks - +5% per rank up to 10, +2% for each rank 11-20, and +1% for ranks 21+; this diminishing return tends to encourage characters to branch out after maxing their key occupational skills; at the high end, attributes provide critical differences since that can add an extra +25% or so regardless of rank. Generally, the same effect could be duplicated (with less use of character sheet space) by increasing the cost of ranks instead of modifying their bonus, although that would give different results on any other rank-based effects (# free levels in 'related' skills?). The diminishing returns occasionally generated undesirable results for melee skills (with difference from level for combat abilities of non-spell-user characters becoming too small).

Skills scaled without regard for dice system: 'The Future Belongs To Us' apparently uses a skill system where skill rating (up to 1000 for some skills) is used to determine what tasks a character can do automatically. This (reportedly) does not tie in much to the general check mechanic of d20 + proficiency score.
Negative skills: HERO 5 (The Ultimate Skill sourcebook) notes that a character can buy 'negative skill levels' - actually a Power, or part of a power -that then applies to opponents' rolls, e.g. combat or for resisting mind control.

Combining level-based skills and 'ranks': the Pathfinder 2e playtest has a level-based skill, plus has 'ranks' that add another bonus i.e. can improve to Master or legendary. An ability to let a character be 'legendary' has a level minimum, so level modifies the check in two separate ways. Its interesting to contrast with Palladium where characters instead normally get large OCC bonuses to raw skill %s - similar in that there's a level bonus and a flat bonuses; but PF2's definition names lock bonuses into later levels in 5E (it seems wrong to have 1st level character be labelled 'legendary' when total bonus is small), whereas in Palladium bonuses are instead always front-loaded.
More often names are just based on a rank (e.g. White Wolf would say 5 dots is 'master'). Or Fireborn list 'primary skills' and 'secondary skills' for each background, which just determines base 'rank' for each - rank 4 for primary and rank 2 for secondary.

Cost of Skills
If buyable in multiple levels, skills may have linear or non-linear costs.
A non-linear cost system in a sense helps balance by adding an incentive to raise lower-rating skills - if your Waraxe-Megadeath-Killing skill is already huge, you might be able to add multiple points of a Craft or Profession skill in place instead of an extra +1 to it, for the same cost. This works as a corrective factor, making it less critical that all skills are equally important/valuable (compared vs., say, d20 System skills that are always +1-per-rank). Non-linear costing can be implemented directly via having costs increase with each skill point, or a similar effect can be a result of the core dice mechanic itself (e.g. for 3d6 roll under, the dice roll's bell curve means later points added increase the total probability by less than earlier points).
Nonlinear costing can occasionally have odd effects e.g. if a game lets a character know one language per Linguistics dot and ranks increase in cost, it takes longer to learn a characters 3rd language than their 2nd (if cost increases but # languages also increases with each dot, cost becomes irregular).
While basically less balanced than nonlinear, the primary advantage of linear costing is that it is simpler. Another compromise to get this effect is to add 'prerequisites' in order to break certain caps - say if raising a skill beyond +10 requires another skill to be likewise raised 1:1, the effect is of a doubling in cost beyond +10. Or a skill raise might require special abilities be purchased separately. The FATE Pyramid is another related, more elaborate example.

Individual skills may all have the same cost, or may have different costs based on the usefulness/specialization of the skill -or IRL difficulty to learn it e.g. in Palladium: powerful skills such as Martial Arts may count as 3 or even 6 skill choices; in 2nd edition D&D, non weapon proficiencies cost between one and three "slots" depending on usefulness of the proficiency. Systems where skills have variable costs rarely have an individually purchasable skill levels as the math is more complex (though examples do exist - HERO, or DC Heroes has a 'factor cost' chart for allowing this even though cost/level is nonlinear). Also note that the game effect of variable cost, and of individually purchaseable levels, is somewhat similar - see 'Modelling Skill Difficulty", below. BESM 3E (and "Dx"?) reportedly have skill costs which are adjusted by game genre. Skill costs -or number of skill points- can also be affected by attributes.

Where skills all have the same cost, skills can sometimes be balanced by having skills that are relatively important (due to coming up frequently, or helping in particularly dangerous tasks) or may be balanced by being divided across a set of narrower skills e.g. instead of one fighting skill individual weapons may be distinct skills. Tasks that are really important are sometimes also split off from what the game formally considers its 'skill' system - for instance a level-based game may have combat abilities directly based off level with no skill involved, or the storyteller game Mage has magic actually use the character's 'Arête' statistic, rather than being a skill.
Systems may have multiple 'pools' of points for buying different sorts of skills e.g. 2E AD&D had separate 'Weapon' and 'Non-Weapon' proficiencies, while Barbarians of Lemuria gives a character separate allotments of combat skills (4 points across Brawl, Melee, Ranged, Defense) and Career points (also 4 points across Alchemist, Assassin, Barbarian, etc.)(Attributes are also generated the same way across Str, Mind, Agility, Appeal).
5E D&D separates out broad 'skills' from 'tool proficiencies' that are much more niche e.g. including musical instruments, gaming sets, or vehicles.

Skill Programs: Modern Palladium games (Ninjas & Superspies, Heroes Unlimited) allows either individual skills to be purchased or allows characters to choose 'skill programs' which are sets of several skills (sometimes, still including some choices). Some of these may form major 'chunks' of classes, with a program sometimes selectable instead of e.g. a major martial arts power (N&S). A program might contain a few good skills, or a larger set of weaker skills. Heroes Unlimited included 'professions' for Alien characters, which became alien 'skill programs' in Aliens Unlimited - aliens had few skills but still these were particularly large, counting as 2 skill programs and with extra skills thrown in if it was the 'profession choice'.
Somewhat similarly, Shadowrun 4E had 'skill groups'; these covered 3-4 skills each, and let a character increase all the skills in the group more cheaply (by defaulting to the skill group score instead), but preventing the character from buying specializations.
Some characters may get an adjustment to skill cost for some skills - e.g. in Exalted based on caste, in oWoD possibly by type [Vampire, Mage, etc.] just as the system evolved from book to book, or class-based in Rolemaster, 3.5 D&D. In generally I would view this as a bad idea (cf. post on point costs).
Champions reportedly had a Martial Artist skill which had a cost equal to a characters STR, i.e 30 Str = 30 points, higher attribute being higher cost.

Skill Improvement Limits: high skill values may be restricted initially by a hard cap for initial characters (can't exceed level X), a soft cap (cost is double above X), a stat-based limit (e.g. Savage Worlds when score > stat costs double), increasing costs that put some values generally out of the reach of starting characters, a limit that a character can only exceed for a couple of skills due to needing an advantage to do so. Original Storyteller had higher skills bought from a smaller budget - skills being limited to *** initially, unless 'freebie points' were used. See 'skill improvement' post below for more on this.

Modelling Skill Difficulty: How difficulty of skill use is modelled also varies between game systems; difficulty modifiers can be 'built in' to the skill score itself, or are expected to be situationally applied. Roll-under systems like RQ set varying base percentages for skills - that is, the base value for a 'routine' skill use factors in that some skills are normally more difficult. Similarly, Synnibarr sets a characters Piloting skill for trained characters at a base of about 100% (before any additional difficulty penalties) so that characters can drive to work without crashing.
On the other hand, D&D sets all skills at the same 1-point-per-skill-point value, and difficulties are instead rated for many tasks, or set by the GM, as target numbers. Consequently while all skills look like they have the same cost superficially, buying 'competency' in some skills where base difficulties are quite high (e.g. Use Magic Device) is much more expensive.
In other words, variable cost of skills, can be replaced by variable 'task DC', if characters are free to spend as many points as they like. That is, brain surgery could be a DC 10 task (but learning brain surgery costs 10 skill slots), or using brain surgery could be a DC 20 task (but your bonus is the same as any other skill). When a player is building a character this gets similar results either way, but how the system works should be understandeable by the player, so they don't under/over specialize their character.
Of the two approaches, variable DCs probably handles untrained skill use more realistically without further kludging; it ideally requires a list of task difficulties more comprehensive than "Easy-10, Moderate-15 Hard-20" however, and caps on skill purchasing can prevent some skills (and hence character concepts) being useful. Variable purchase costs mean characters must buy 'competency' as defined by the system to get the skill (no medical school washouts or semi-competent rocket repairers!) and can't break the skill-bonus system, but the skill purchase system becomes more complex (i.e. probably a longer list of detailed skills).

Attributes & Skills
Frequently, skills are modified by one or more attributes. Some games limit this to one (sometimes which one varying from occasion to occasion), others apply more e.g. Runequest often adds two or three. Keeping skills with the same # attributes keeps it perhaps more consistently scaled, though its perhaps not unrealistic that people vary more in capability with certain things. Games can sometimes have skills which have no stat modifier (again often realistic in context, but dependent on what stats the designer has already decided to include).
Attributes generally modify skills by either:
 
*adding a bonus to the skill roll (either a flat number or extra dice).
Elaborations here - systems may allow for cross-matching a skill to a different attribute depending on circumstances (e.g. the Storyteller system, or the original Cortex);or via advantages/feats letting characters switch controlling attributes (which works as long as attributes have a fairly consistent scale i.e. some can't rate twice as high as others, albeit that it may lead to different-cost i.e. unfair methods of maxing out skills).  (See also 'Weapon Proficiencies' later for weapon-based adjustments). Some situations might penalize characters by preventing them applying an attribute bonus, e.g. limited mobility preventing a character from adding DEX to attack.
Mutazoids has multiple possible defaults for some skills, but with some being lower i.e. Administration is either Cha/3 or Int/4.
Systems may also unhook skills from attributes for certain characters, e.g. FATE variant 'Bounty Hunters of the Atomic Wastelands' often has Skill ratings equal to attribute score, or a special ability can disconnect it/make it a step higher if the attribute is maxed fully: e.g. "Keen Senses: a Bounty Hunter's perception is treated as Great regardless of his Thinking score. If his Thinking is already Great his perception becomes Superb"; similarly an early D&D Next playtest let thieves default to a stat mod of +3 for some skills.
Quite often games will double stat for raw stat checks (e.g Unisystem, Magic Quest, Talislanta). Done well this lets the same difficulty scale apply for both raw attribute and [attribute+skill] checks. Making this work well is one argument for having large stat numbers (e.g. 1-10) instead of just modifiers rated around average 0 with correspondingly lower target numbers.
Some skills as noted above may lack a controlling attribute. If stats are random (particularly if adding stat directly), it might work to add extra dice as if generating an attribute to each skill roll (i.e. stat rolled on 3d6+skill 2d6 - a bit like Tunnels and Trolls? - just roll 5d6 for skill). Or oWoD Storyteller could use a Luck Roll's successes l to generate a # dice if no attribute applies, then add skill dice and reroll.

*being rolled for skill checks directly (i.e. Warhammer 1e/2e, 2nd Ed D&D Non Weapon Proficiencies/ Basic D&D General Skills). In these cases what to roll for 'untrained' use is generally vague (2E) or defaults to e.g. 1/2 stat (WHFR 2). (And World of Dungeons as noted above has no bonus for having the skill at all, but worsens failure results for untrained use).
Hackmaster often has skills based off two attributes, with the lower of the pair used unless an extra 'build point' is spent. Beyond the Wall (reportedly) uses stat rolls and adds a +2 to a stat check for a relevant skill, which (if PCs generally pick skills for stats they're also good at) makes the die roll nearly superfluous (near 100% success rate).
This is possibly useful in allowing a skill to use multiple stats simultaneously, by averaging the stats. Forgotten Futures is interesting here in often using avg. of multiple stats, despite having only 3 stats - Body, Mind, Soul.

*modifying skill cost (usually assumes skills are purchased in multiple levels). E.g. in Savage Worlds, an attribute's main effect is capping when a skill goes to double cost. As Edges and skills are bought from a single points pool, a low attribute can thus give an incentive to pick up an Edge that would normally be suboptimal instead of raising a skill further.
A few systems (WEG's Star Wars, Feng Shui) give skills non-linear costs based on total attribute+skill, i.e. a high attribute raises purchase cost. JAGS and DC Heroes provide the option of "linking" a skill to an attribute, or having a wholly separate skill score.
The downside to this approach is with regard to balance - for example, a high stat may or may not be a benefit depending on which skills you pick. ( See post #121 on point buy variable costs for related discussion).  
A 'cost' modification can also be done by first assigning points, then adding a bonus to them based on attribute; so, Basic or AD&Ds' 'bonus XP for high prime requisite' is another version of this.

*modifying improvement rolls (if skill gains are random). HarnMaster has a 'skill base' calculated by averaging up to three attributes, with improvement rolls getting a bonus for high stat (see next post).  
 
*determining the number of skills (or skill points) a character gets initially. This is fairly clumsy and awkward compared to modifying the purchase costs of skills for the relevant attribute - it has a similar overall effect (i.e. high stats = more skills), but makes it much more awkward to represent how different attributes give benefits to different skills.
Synnibarr and SuperWorld both add together a set of attributes to give a total number of "skill points" to spend on skills, meaning a high STR can give you the points you need to learn Rocket Science (...or for Synnibarr an extra cool mutation) ; Buck Rogers High Adventure Cliffhanger game gave 4 skills to the top-ranked attribute, 3 to the next higher rank, and so on down to 1 skill for the character's worst stat.
 
A couple of games have stats modifying per-level skill points as well. Rolemaster has 'development points' based off a characters several random-rolled statistics, while 3E D&D has a simplified version of this modifying points by INT modifier each level. With random roll this can have unfortunate effects, with higher stats continually drawing ahead (though RMs' diminishing returns on bonus from rank may slightly ameliorate this). (HARP had a particularly bad system for this where stats determined development points, but development points could then also be spent to raise stats in a feedback loop).
 
*a skill check may default to an attribute check if the roll fails (so the skill gives an extra separate roll). For instance, in Gangbusters a character with Boxing can roll their Boxing skill to hit an opponent; if this fails they receive the normal Agility check to do so (success on the boxing check indicates two hits, rather than one).
 
*default to attribute check if no skill, otherwise unmodified skill check (DC Heroes, Shadowrun 1e-3e: see following section). This relies on players' being aware of the system and building around the 'trap' option, as a character with low skill may be better off defaulting to attribute anyway.
HERO [4E] uses the opposite system - background skills default to a base chance, but can be upgraded to a 'characteristic roll' for a character point if desired.

*skill and attribute may do different things e.g. in Legend of the Five Rings, skill determines number of dice rolled and attribute the number of those (starting from the highest) that may be kept and added together. In Wolsung (see reference here: http://forum.rpg.net/showthread.php?628579-Interesting-Attrribute-Skill-interactions ) a d10 system is used where skill increases the threshold at which dice explode (e.g. Strength 2 higher than normal allows a reroll on 8-10, instead of only 10) while Skills just give a flat bonus. Arrowflight 1E has dice pools where stat determines number of dice rolled and skill determines target number.
 
*'Edge of Midnight' has all tasks using separate [1d10+trait] rolls for skill and attribute, with effects on how the success is narrated (i.e. complete success if both succeed, total failure if both fail, partial success by skill, or partial success by raw talent).

*occasionally (Palladium; FUDGE) attributes may have more or less no effect on skill scores. (Fudge evolved into FATE, using Aspects to plug this; Palladium actually does it in reverse, with skills modifying attributes instead of the other around e.g. "Running - +1 to Endurance and +4d4 to Speed"). To an extent I suppose Palladium's system is justifiable if it is assumed that a character develops particular talent in aspects of an attribute related to their skill (climbing muscles for the guy who climbs alot, strong right arm for a baseball pitcher, etc) which then overflows into their general attribute). FUDGE does suggest that characters stats and skills be built logically by the player i.e. there's some expectation that the player might assign good physical skills to a character with good physical stats.
 
*potentially, in a system where attribute modifies advantages/disadvantages (cf. ROAR where a high stat gave 'faculties' and a low stat gave 'flaws' e.g. a low CON making a character a haemophiliac, low STR dwarfism or low CHA scars or exile) attribute modifier could be dropped entirely to solely use ad hoc modifiers from advantages/disadvantages. You can again compare this approach to say 0D&D, where different (but not game-mechanically-defined) ideas of how a stat works could affect checks, such as a character with Dex 3 being 'as blind as a bat'. This approach also operates slightly as a secondary mechanism for adjusting checks in Savage Worlds, where a particular stat may be a prerequisite for an Edge (Spirit d8 to get Charismatic, for example).

*Other weird: Fireborn is interesting in that skills are buyable in multiple levels, but are basically a raw attribute check. What skill ratings do is that characters can move dice from another stat onto the main stat roll, and this is capped by the skill. (Perhaps its almost-binary since the system can't account for different skill ratings in action sequences, see 'actions' post). Higher skills' then give extra dice, but generally at a cost e.g. more melee means higher attack but less defense (the exception being non-combat scenes where the reduced stat likely won't come into play). Skill rating also limits how many associated manuevers can be piled into an attack sequence e.g. a character with Athletics 3 could [Climb+Climb+Climb] but if they had 5 actions, couldn't spend the last two climbing.

As noted under attribute modifiers above, skill and attribute vary considerably in their relative weighting on tasks, due to how much of each contributes and the relevant variation in each (...if everyone has an 18 for a +4 bonus, its real 'weight' is limited). Skill labels are often added to skills (Whitewolfs' 5 = master for instance) but this labelling can be deceptive if attribute weighting is high.
Relative weighting of bonus from attribute, vs. bonus from skill is something to consider carefully especially if you intend to allow purpose of attributes and skills from the same pool of points instead of suballocating pools, since it may be more optimal to spend points on one over the other, like how in GURPS its usually preferable to go for attributes. In any system watch relative stat/skill gain costs from earning XP or levelling, which is usually a common pool and may have problems too.

Overall: Attribute modifiers can boost low/untrained skills particularly, boost high-end skills particularly, or boost everyone equally. Which depends on method chosen, and probability curve of the core mechanic itself e.g.
Savage Worlds [High-end] - stat raises 1:1 purchase limit.
Storyteller- [All Equal] - albeit for many checks if 1 success is all you need, the stat helps low skill ratings more.
D20 [Low-end] - bonus outweighed by level bonus at high level.
Star Wars D6 [Low-end] -increase cost for skills is based on total score.
HERO [Midrange]- low values increase more due to 3d6 curve, then hits scores cap.
Overall character-benefit of a high score (for skills based on it) - will depend on its cost, and how common low or high skill ratings are.

Talents
Game systems can also try to give characters skill modifiers from innate ability separate to those defined by the games core attributes.
This may be for more realism for instance or because the game has a limited number of attributes (or no attributes), to supplement attribute modifiers where the effect of those is low, or add more variation to skills if these are fairly fixed. Subabilities, feats or merits/flaws can do this, or 'talents' might be determined randomly, either in a generic or very specific case-by-case fashion. Skill specialties (or just higher skill purchase) could also represent talent, if the game system doesn't want to differentiate between innate and learned skill boosts.
Fantasy Games Unlimited's range of RPGs (e.g. Daredevil, Aftermath) included a set of modifiers to skills from "Talents", each of which covered a skill category, as well as attributes. Dice rolls generated base talent ratings from -4 to +4, with a pool of discretionary points also added. Talents could also provide special advantages (but are presumably not usually tested individually, the way attributes are). deadEarth had characters roll for each skill for inability/ability (it uses an Xd6 dice pool with ability letting 1s be rerolled and inability makinig 6s reroll once - abilities and inabilities are by default rare -2 or 12 on 2d6 - but the mechanic is also used by mutations which may give particular abilities or inabilities). Synnibarr has an optional roll to determine if a character's learning rate for skills is modified. GURPS has a number of specific advantages/disadvantages; it showcases one problem with these, that being skill boosting merits mainly just create ways to alter effective cost of skills (and create less balanced characters) in context of its point-buy system, a problem difficult to fix when skill costs are non-linear.

Other subsystem modifiers to Skills:
Not infrequently advantages/disadvantages may modify skills as well as raw skill point expenditure.
Savage Worlds has a lot of these, for instance. FantasyCraft is interesting in having feats that alter 'crit range' of skills (raw bonus doesn't affect this).

Skill categories & specialties
Some skill systems also include rules for specializations in skills e.g. Shadowrun. Similarly, LegendQuest has multiple skills of varying widths which are potentially stackeable onto the same roll - "group skills" including Melee and Rogue and potentially fulfil the same sort of role as 'classes' by giving basic ability in a set of skills, including cancelling the default -15% untrained penalty. (Kerberos Club for ORE has a similar arrangement with Broad/Flexible/Influential skills, with broad skills useable in various profession-related ways and likewise multiple skills potentially stackable). CORPS has (reportedly) stackable broad Primary skills, then Secondary and Tertiary skills - secondary rating can't be higher than Primary and Tertiary no more than half secondary.

DC Heroes lists various 'subskills' for each skill; a character can buy just one or two subskills if desired, with a cost discount [Though being good at particularly niche things can also be the "Scholar" advantage, which is more expensive, e.g. Lobo is a 'Biology' scholar with some scientist skill]. Most specialization rules reduce cost to buy higher ratings or add a bonus, though deadEarth has specialization skills used to roll against a lower difficulty (useful though it doesn't help with opposed rolls). Another variant for a dice-pool-step-die system might be to have narrower specialties have larger dice for tasks.
Specialties might be purchased with skill points as skills are, or, a skill might give a free 'specialty' when chosen (other related uses being penalized), or a specialty might be gained automatically at a higher rating - oWoD Storyteller gives a free specialty (reroll of any 10s) to any skill at 4 dots, while HarnMaster lets a character get a specialty (+10%) in a skill at 70% and again each further 10%.
Freerpg FURPIG lets characters who reach 'perfect mastery' of a skill or 100% (limited to one character per skill in the universe) create new super-skills building on their mastered skill.
Alternity had 'broad' skills which were yes/no - without the broad skill a stat check to use the skill was halved - then "specialties" which added a bonus to more specific actions (e.g. broad skill Athletics, specialty Jump). It had rank benefits for higher specialty ratings, with characters also sometimes being allowed to spend skill points to gain a benefit prematurely e.g. brawl 8 gave a damage bonus with unarmed attacks, but a character could spend skill points - based off the rank difference - to get this earlier).
Hero has 'penalty skill levels' letting a character purchase reductions to situational penalties, such as range modifiers, called shot penalties, throwing unbalanced objects, or being prone/encumbered/underwater. (As well as possibly 'rapid use' penalties). Some powers or advantages could also modify skill use (e.g. "A character who buys his legs and/or mouth as Extra Limbs suffers no penalties for performing Agility Skills with them" - The Ultimate Skill supplement for Hero).
Original Cortex reportedly had broad skills which could be improved up to a threshold; beyond that a character had to choose and individually improve specializations.
Skill specialties are handy for adding depth to characters while still having very broad skills, and thus a short skill list (a normal problem with broad skills being that characters can end up with odd competencies due to skill conflation, like Pathfinder paladins being good at Gather Information (a Diplomacy function in PF).
5E D&D has very broad skills  at a quite low bonus - being trained in a skill at 1st level is a +2 bonus or +10% - with 'subskills' generated via specific class/race abilities -e.g. rangers double their proficiency bonus in a particular terrain, while dwarves and gnomes double their History skill with stonecunning or 'artificer's lore' respectively.
In a sense raising attributes has a similar effect to having 'broad' skills in most systems though even broader (assuming a +attribute bonus to skill checks, or default to attribute - it doesn't apply in e.g. Savage Worlds, and barely in Runequest).
Skill specialties can overlap somewhat with "complementary skills" (see later). For example, Mutazoids would let a character roll a Jury-Rig skill check to get a bonus on a Mechanics skill roll; Storyteller would instead have a mechanics-type skill with a 'jury-rig' specialty. Mutazoids here has the advantage of being able to define different levels of jury-rig skill (not just yes/no; although, other games do have specialties that aren't just binary) and allowing matching of Jury-rig with other skills if need be (though that might not really be required either, the principle could apply to other cases).

Skill-based special abilities
A simple case of this is where some threshold lets a character skip a check e.g. oWoD vampire lets Drive * handle automatic vehicles and Drive ** for manuals. That can sometimes happen automatically as a result of e.g. 'take 10' type rules [cf. post 28, excess rolling].
4E D&D (in PHB-3 and Dragon magazine) has "skill powers" which are available to characters trained in a skill, and which can be taken instead of a normal (class) power when levelling up, giving extra abilities related to a skill (like 'fast hands' for thievery, or ability to substitute Knowledge checks for raw Int checks). These count as alternate 'utility' powers which are only gained at e.g. level 2, 6,10 etc - fairly artificial and they can't be used to make more diverse or skill-focussed starting characters.
3E D&D (in Complete Adventurer) had "skill tricks" which cost skill points but which worked similarly, giving characters special abilities for skills like 'whip climbing'. A game with advantages can similarly limit access to some of these to characters with an appropriate skill (no doubt GURPS has done this at some point). Effect-based systems (see the super powers section) can model this backwards sometimes - M&M (I think) allows characters to take a 'requires skill roll' as a limitation on a power to represent something being skill-based which by default needn't be, although this is problematic since buying enough skill ranks makes the actual detriment of the limitation questionable.

Skill Lists
Skill Lists may be described as edge-defined or centre-defined.
(see thread: http://www.therpgsite.com/showthread.php?t=18250 )
 
Skills are sometimes player-defined (e.g. in Teenagers from Outer Space); in other systems basic skills may be defined but characters may have some leeway in inventing specialties (e.g. Storyteller).
Length of skill lists: Length of a skill list usually flows from other concerns (i.e. definitions of the individual skills) though Fate Core attempts to work the other way, with length of the list chosen to generate enough diversity between PCs while still giving parties access to most things. Skill lists are also sometimes rejiggered to balance out number of skills relating to each stat, and hence relative stat importance.

Games can have shorter skill lists if 'class' normally covers that purpose, or if characters are largely differentiated in other ways, for instance in a supers game where powers largely differentiate characters:  early Champions for instance was said to have had a fairly simple skill list until the HERO system started to cover the non-supers genre with other games, and many others e.g. MSH likewise gave out only a handful of simple binary Talents giving +1 step to stat checks. Overall simpler games also have simpler skill systems. Another interesting case is drama-based Cortex+ game Smallville, where personality/relationships somewhat assume the role of skills for checks.
There is some discussion here on the merits of 'narrow' vs. broad skills (broad being the current trend, with PF/4E and 5E D&D having skills designed to more or less cover every possible eventuality).
Skill 'tiers' (see below) also make for longer skill lists - (e.g. one Mathematics skill vs. separate 'Basic Mathematics' & 'Advanced Mathematics' skills).
Games with fairly vague skill definitions, leave the GM some leeway in deciding how 'broad' a skill list is. As a weird example Gamma World D20 is actually fairly defined (=d20 modern), but also claims in the GM guide that the GM can or should alter skill 'breadth', e.g. whether Craft skills also include "related bits of lore" depending on if they want a pulp hero feel vs. gritty feel in the campaign. Fiddling around with this significantly also alters the balance of power between classes that are largely skill-based vs. those that aren't.
One mechanical factor affecting length of a skill list may be if skills automatically scale (e.g. with level) or not: having skills scale up automatically is more compatible with a longer skill list (e.g. Palladium), whereas allocating points is more likely to be unbalanced the more skills there are (it becomes hard to keep up with all expected skill requirements and generates significant areas of incompetence for characters - maybe 3E D&D).

Skill tiers
Skills may form tiers where one skill is required to buy another skill. Palladium uses this a fair bit (i.e. Math-basic and Math-advanced are separate skills, instead of Advanced being the same skill at a higher rating). An approach such as this can let characters be built that can achieve specialized tasks easily without being generally competent in other areas i.e. the rocket scientist who has Rocket engineering, vs. the rocket scientist with Mechanical engineering at super-high levels and can also fix anything else ( although with a more detailed skill system required compared to just having 'specialization' rules).
Skill tiers in this case may be a consequence of binary skills e.g. for Palladium while characters have varying %s, purchase is binary so you can't choose to be good at something, except by buying multiple distinct skills (apart from OCC bonuses to skills). Dark Heresy may be another example of this.
Lords of Creation built tiers into skills explicitly, with each rank of a skill group being a new skill - which did mean characters couldn't be good at some skills without learning vaguely related skills. deadEarth had complex prerequisites which sometimes accidentally formed 'trees' - Armor Repair requires Smithing, Construction and Chemistry, Smithing in turn required Metallurgy and a Weapon Skill, resulting in some skills having large point costs to buy. (everyone has herb lore, but Haggle is rare due to having 5 prerequisites; Intimidation requires Resolve which requires Running).
 
The free rpg JAGS has an odd rule which seems something like skill tiers; characters buy a skill level for rolling against on checks separate to a rank description ('level') of the skill - Beginner, Professional, Expert, or Master. A character suffers a -3 per level beyond their own, and usually a discount on difficulty penalties, but a character with a high score may be able to do routine low-level tasks more reliably. This sometimes represents changing tiers of skills (Level 3 Lockpicking is 'safecracking') but its not always clearly designed.
 
Skills which 'overlap' sometimes use different task difficulties for the same task. Risus for example describes this as 'subjective difficulty': tasks become harder the further removed they are from a character's "cliche" (the Jungle Lord can swing on vines automatically, the Swashbuckler easily, while other types find this more difficult). A more formal version of this would be skill defaults (below).
Risus also very generously allows characters to substitute "inappropriate" ratings for checks as long as the player can describe it, and rewards this by giving "inappropriate" ratings additional damage in contests - i.e skill misuse is encouraged for the humour value.

Skill defaults and untrained penalties: [attribute+skill] systems generally default to just [attribute] for skill checks. However, a key concern is often to give a character with some training a hefty bonus for knowing a skill vs. the character with no training hence additional penalties may apply. If a game uses skills for combat the decision for this will also obviously impact untrained weapon use.
Some examples of default systems:
*In oWoD a character rolls [stat+skill] but the target number may also increase if the character has zero skill, or a check may not be allowed.
*A few systems may have base ratings for all skills e.g. BRP/CoC/RuneQuest - lack of an easy default system in BRP means it needs fairly long list of skills to cover most eventualities; a raw attribute check would have a much higher chance of success than a skill check. Here there are no extra penalties for not having a skill, but 'competency' represents a large investment of skill points for many skills.
*D20 system: this gives characters x4 skill points initially (so a 'starting' skill level is +4). Pathfinder instead gives any trained class skill a +3 extra bonus.
Normal 3.x has some characters unable to use some skills untrained, or for some skills just not use certain functions (others are fine). FantasyCraft by comparison has a DC-based ban (if untrained, can't pass DC >15) instead of specifically task-determined. IIRC.
*The D&D 5E playtest of midyear 2013 added 'skill dice' to a d20 roll i.e. a roll is d20 + stat modifier, +d6 if the skill is trained. This gave diminishing returns on skill points spent as the largest bump is at the initial skill level (+3.5), while successive improvements in skill to d8,d10, or d12 have less effect (+1 on average each time).
*GURPS has a complex system where each skill lists several other skills that can be substituted at varying penalties - a character might be able to attempt a combat move using Dex-6, Karate-3 or Quarterstaff -2, or something. Metascape similarly has varying penalties based on how similar a check is, but penalties are GM-assigned; it also recommends the GM reduce penalties or make tasks easier if there are no skilled characters.
*Shadowrun 1E has a complex flowchart, the "Skill Web", indicating how many "steps" any skill is from any other skill or attribute (=what penalty), which fulfills the same function as GURPS defaults. This could make defaulting a better option for characters with low skill and a high related skill or stat, depending on target number and how many successes a task required, but with which to use being fairly non-transparent. Characters usually got no bonus from stats to skill checks in 1E unless using them as the default (although in combat a character might get extra dice from 'action pool').
Shadowrun 3E replaced the Skill Web with a default rule that if a character with no skill could roll attribute with a +4 target number penalty; 4E dropped this as TN was fixed at 5 and as rolls went from just skill in dice to [skill+attribute] in any case, but had an extra -1die untrained penalty.
*Default to stat roll. DC Heroes does this; characters either use a bought skill value with no attribute bonus, or default to [attribute score] with a column shift penalty. It has the problem that characters with very high attributes should not buy skills, or only if they buy large amounts of them. DCH characters can buy points in a skill equal to the attribute exactly, no higher or lower, as a 'linked' skill, with a cost discount, but overall costs are fairly ad hoc.
*Palladium (according to one Rifter) suggests d100 under stat instead of the normal percentage which gives very low default chance -unrelated to how difficult the skill normally is to perform, which sets the base percentage.
*Skill-based dice pools with low attribute mod: here the task mechanic is very deterministic based on skill level, inherently making untrained rolls difficult or impossible anyway. For instance, Central Casting suggests a skill system where a character rolls d6s equal to skill level (additive), plus or minus d6 for a high or low attribute.
*Default to 1/2 stat instead of full stat: often used where a skill would ordinarily be a stat check e.g. Amazing Engine, Dark Heresy/Deathwatch. Note that '1/2 Stat' gives a very different probability depending on whether a game uses a linear die like d20, and multiple dice like 3d6 roll under (much less than half).
*ZeFRS (originally in the Conan RPG from TSR) - here there are no attributes but a characters total ranks in groups of related skills ("talent pools") are added together and divided by 10 to give a default talent rank. The idea is interesting but this requires a fair amount of granularity (ZeFRS uses a d100 table lookup), with base chance of success being quite high even for zero talent; while being more realistic in some ways than say GURPS despite the latter's complexity, the bonuses involved are small and fiddly (unlikely to be more than +6% or so).
*D&D 4th edition has binary skills where a character gets +5 if trained in a skill. Characters can take a 'jack of all trades' feat giving +2 on untrained checks (i.e. half the trained bonus).
*Talislanta rolls d20+stat+skill level, but uses d10 instead of d20 for untrained rolls.
*Savage Worlds uses a default roll of d4-2, and the -2 also applies to the Wild Die (normally a d6 reroll with no other adjustment). It also allows a "Common Knowledge" [Int] roll for areas not really covered by skills, with modifiers for character background. As noted above, stat affects purchase limit for skills, but doesn't modify untrained skills.
*LegendQuest has a fixed penalty (-15% for untrained use), but with the interesting idea of 'cross-matching' untrained penalties i.e. attempting to use archery from horseback without Ride applies the Ride unfamiliarity penalty to bow use.
*The GDW House system (Twilight 2000 2nd ed, Cadillacs and Dinosaurs, Dark Conspiracy) is a skill-based system with no defaults. Skills are bought up from zero and an easy task by default gets 2x rating; hence characters are by RAW often unable to attempt many basic tasks unless they've put ranks in them including shooting guns, shooting bows, Perception checks, and Persuasion checks. The skill system covers most tasks quite comprehensively, which makes the problem with Perception harder to houserule fairly (adding a default rule likely disadvantages those who took the skill, whereas if no one had Perception skills it wouldn't). T2000 edition 2.2 of the rules changes to a D20 roll under [stat+skill] to address issues such as this. Cadillacs & Dinsosaurs does often give a couple of ranks in skills for free based on background, including shooting - the "free skill" sort of works for shooting in that some characters (e.g. the Wassoon tribe) are quite primitive and wouldn't have any knowledge of firearms, though its a little too easy to accidentally create characters unable to fire a gun. The 'free skill' approach does make maxing-out those skills more expensive than raising the benchmark performance for untrained characters would have.
HERO similarly has 'everyman' skills, where every PC may pick up some specific skills. Giving everyone a skill is probably less useful than adjusting default difficulty and whether 'untrained' use is allowed.  
Giving out 'free' skills could sometimes be an easy way to generate some base values or fudge numbers; for instance, if everyone has a few ranks of 'Running' skill or 'Body Development' skill, this could be used to generate movement rate or HPs without overcomplicating a formula (having to add an extra base value) and while keeping maximum divergence in ratings low.
*The HDL system uses (basically) stat+stat for stat checks and stat+skill for skill checks. The 'good looking' merit makes a seduction roll from a skill check to a stat check (so that the merit effectively gives free skill points equal to stat number). Its interesting, though this has a couple of problems: free point shortcuts like this can create multiple ways to build a character some of which are cheaper, and the merit is conceptually redundant with the games' actual Looks attribute or at least poorly named.
*in 2E AD&D, skills (NWPs) have no defaults, but what skills cover isn't well defined and untrained use isn't expressly prohibited so, GM-permitting, an untrained character may still be able to pull off the equivalent of successful "skill use" with an 'old school' approach, getting the GM to describe the situation exactly and reacting with player skill.
*In Warhammer 2E characters typically had a skill or didn't (rarely it may get a +10% or +20%); WHFR-4E instead distributes 'advances' that are +1% increases. Similar to 3E not all skills can be used untrained and a character needs to put 1% into an 'advanced' skill to use it, but 4E looks more annoying in this regard in that the points are more more fine-grained. Which is to say, trap options exist in that a character can choose between a +1% that increases their stabbing skill from 30% to 31%, and a +1% that lets them use e.g. a Lore skill at all (from 0% to 31%).
 
A couple of games have skill 'defaults' but still let characters drop below default - e.g. 'The Agency' (as mentioned in Attribute Effects), or Genre Division 3 (reportedly) where characters can have 'incompetencies' rated at -1. Original FUDGE also usually defaulted skills to 'Poor', but with characters able to reduce some to Terrible in exchange for a skill rank elsewhere. Games with disadvantages can also have disadds that ruin particular skills e.g. 'All Thumbs' in Savage Worlds. Whether this is really necessary depends a bit on how competent having any ranks in a skill makes the user.

Special: a couple of systems have no specifically defined mechanic for skill use. A primary example would be Secondary Skills (from the 1E AD&D DMG, but most often recognized from 2nd Edition D&D) which describes vaguely what a character is capable of but leaves the exact mechanic to us (ability check, GM-defined percentage, saving throw, etc.) at GM discretion. Similarly Superbabes defines how to resolve some specific cases (usually with d20 roll under ability checks) but has no defined method for using skills overall.
Reputedly, Traveller also originally used different resolution systems for each skill.
 
Extended Checks: A few systems (Alternity, 4E D&D, Shadowrun, White Wolf) have skill mechanics for complex or extended skill checks - see more in future sections dealing with e.g. Crafting. These tend to work most easily with dice pools. An extended system tends to be used in RPGs for representing lengthy actions, although they are better really for cases where a single die roll is too polar, where multiple characters can help/cooperate, or possibly (it is argued) when useful for building dramatic tension.
More successful versions of this give some way to have a higher success grade on a single check make more progress e.g. by counting successes (dice pool); Alternity had a system where a good success (under 1/2 skill) counted as 2 successes and an amazing (under 1/4 skill) counted as 3.
 
Skill-less systems (see also post #21, derived attributes): older versions of Tunnels and Trolls have no skill system; rarely in play a character may get to roll to see if they have a particular skill (e.g. an IQ roll to see if the character can play a piccolo when fighting a Shoggoth). It does have a language system.
0D&D/ early 1st edition AD&D has no skill system, with class determining most of what a character can do and player skill often used to resolve other actions (i.e. describe to the GM how you would do something and they determine the result).
Gamma World (4E) gives skill points to spread among 'class skills' but handles most tasks with derived attributes, except for a % by class to know Reading/writing, Riding and Swimming.
Maelstrom mostly defaults to pure attribute checks with occupation giving some special abilities. While there are no general skill points, 'preaching' ability is based off # of years spent studying the ability.
Skill-list systems can also require GM arbitration of tasks based on character background to determine what a PC can do.
 
Complementary Skills: FUDGE lets a character with a (GM-determined) secondary trait add a +1 to the task if Good or better. Fuzion lets a character roll one skill in order to get a +2 to another skill roll (e.g. using a successful Fashion roll to get a +2 on a follow-up roll to Seduce) again GM-determined. This rule can be used to generate e.g. attack bonuses for successful stealth rolls and the like. (Note that this system where a roll not an extra skill rank adds a bonus, a character might be able to leverage extra benefit from a high attribute, and/or involve extra attributes - e.g. in the above example, both Fashion and Seduce might be Cha-based, letting the character who passes the synergy roll due to CHA effectively 'dou

The "Play Rifts" skill starts at 15%, but you get +7% per level.
So by level 8 you can be successfully playing Rifts fully 50% of the time."
-OptimusZed, penny arcade forums.

Skill improvement resources in many systems are linked to other character improvements e.g. based off level, or points may be spent on either skills or other improvements.
 
*RQ/HarnMaster usage-based improvement- significant skill use gives a character a “tick”; they roll over the skill (i.e. the skill check must be failed) to get additional skill points; this slows down increases in skill values as the score increases (conversely to this, Synnibarr gives improvement for a skill only if it is used successfully and subtracts if a character fails checks, meaning high skills get higher and low skills get lower (!)). RQ doesn't handle gaining advantages/disadvantages etc. as RQ doesn't traditionally have a major advantage/disadvantage system; some attributes in RQ may be boosted with training. IIRC RQ doesn't modify improvement rolls for high attributes, so benefits from these erode over time; HarnMaster however uses a roll of d100+ Skill base (the average of three 3-18 controlling attributes for the skill) so that higher attribute characters keep ahead on relevant skills.
Risus likewise has usage-based improvement - a "cliche" rating improves if all its dice are rolled and come up an even number, so again higher scores are less likely to improve.
Skyrealms of Jorune is noted here as similarly having rolls to improve skills, however, rather than being vs. the skill score, a roll is made against a characters' "Learn" attribute (linearly rolled!), so characters with a high stat have a permanently much faster advancement rate.
Gamma World 5E has a usage-based system for psionics (each time you use a power, gain +1, a character can spend bonuses to try to learn a new power), despite other advancement being XP/level based.
Mercenaries, Spies and Private Eyes has characters earn XP separately in each skill they know, e.g. 50 per round used; this uses the same table as character advancement in level, but that earns xp more slowly. New skills can be learnt only by raising the Intelligence attribute, which gives more skill points.

*time-based e.g. the ALIENS Adventure Game (Leading Edge Games) gives a skill roll to improve after each year of game time. StarCluster adds a skill improvement each year at the cost of an attribute decrease (Immortal characters in Outremor get neither, but can shift a skill point around).
 
*Training/money may be used to purchase skills, rather than experience points (RQ again; SenZar; Languages in AD&D (if not using NWPs) or in Dragon Warriors.
 
* the GM may award ad-hoc increases. Similarly to this, Amber uses points, but with the GM tracking advancement of the player without their knowledge of the PCs exact capabilities.
A sort of example of this- although not a skill improvement per se - may be fairly obscure freerpg FuRPiG, where the GM is allowed to give permanent "Bonus Hit Points" to characters who "have been particularly valiant in battle" and "continue to fight against extreme odds right up to Death's door". In similar vein, some Tunnels and Trolls adventures would grant stat raises for with little or no explanation ("you spend several hours conversing with Lorac...add 3 to your IQ for absorbing some of the much and varied knowledge of this man"), and in other places grant bonus skills for some event (visiting the Archer's Guild in City of Terrors, making an IQ SR to know how to play a piccolo in Arena of Khazan) - the Deluxe T&T rules formally recognize this, giving the GM ability to reward PCs with new 'Talents'. In many systems ad-hoc GM adjudication handles fuzzier cases outside the main rules e.g. gaining reputation and the like - though GURPS would probably charge characters points for adding the "Welcomed as a hero in Turdhaven" Advantage after saving the town, of course.
 
*XP: Characters may be awarded XP, which are spent to either increase the characters’ Level, or to buy up specific skills. (Talislanta allows XP to be spent either on putting up Level – which increases all a characters career-related skills - or buying up specific skills individually; among other things this meant the GM can stat up a generic NPC of a high level quickly, while still giving PCs using the same system the flexibility to add skills beyond their character class).
In a level-based game, there might be have abilities based off exact XP total as well i.e. in Basic D&D demihumans eventually would cease levelling/getting hit dice, but still would gain combat abilities at specific xp totals; 1E AD&D Barbarians gained a 'Horde' with 1 member per 1000 of their XP. Savage Worlds gives "advances" (a new skill, pair of skill increases, Edge, or stat bump) every 5 xp and a full Rank increase every 20 xp (which mainly gives them access to new and better powers or Edges).
Skill-based systems that use XP (but not levels) may still have other limits to how far skills can be improved. Limits can be attribute based (Savage Worlds); FATE has a 'skill pyramid' whereby a character can only have so many skills of a certain rank without having more skills at a lower rank. This occasionally annoys people ('I can't improve my Drive skill until I've gained a new skill in Interpretive Dance') (perhaps letting players create new subskills would help here, so they buy a separate 'Handbrake Turn' skill instead).

XP or equivalent rewards may be awarded for various actions including defeating monsters, delivering the final blow on a monster (MERP), bravery, surviving, clever ideas, avoiding unnecessary violence (Palladium), roleplaying a character well, self-sacrifice, annoying other PCs (picaro rules, with thanks to Shipyard Locked), rolling critical successes, rolling any saving roll with an amount based on dice roll/level of roll (Tunnels and Trolls), failing rolls (Dungeon World), travelling a certain distance (MERP), just attending sessions (Storyteller), keeping diaries or writing backgrounds for characters, helping other players (particularly newbies), making the GM laugh, making magical items or furthering their god's ethos, performing other class-related actions (2E D&D), finding treasure (Basic D&D), or squandering all of your treasure between sessions (Barbarians of Lemuria). Extra bonuses are sometimes awarded to lower-level characters (3E D&D) or characters with high ability scores; AD&D gives XP bonuses for a high 'prime requisite', balancing classes with more powers by giving these multiple prime requisites (or no prime requisite) to make it harder to gain an XP bonus (XP bonuses for high stats started in 0D&D, where stats gave few other modifiers - at most +/-1). An otherwise largely useless stat, e.g. Charisma in D&D clone 'Dangers & Dweomers' can also be balanced by giving XP modifiers. XP penalties may also be applied to characters of certain races or race/class combinations. XP is usually GM awarded; Teenagers from Outer Space suggests a XP ballot where all players vote.
 
AD&D multiclass characters had separate XP pools for each class (one early AD&D class, Entertainer in Dragon #69, has three separate sub-classes that can advance separately, with two needed to improve to 10th level and three to 11th, plus optional magic-use and complicated HP determination system); a system could also theoretically have multiple XP pools for different things (roleplaying vs. combat).  Fireborn has characters that are humans reincarnated from dragons, and so advancement points are divided between 'humanity points' and 'heritage points', given by the GM based on the nature of characters' adventures, but can trade between types at 2:1 (this sort of thing could also be patched on fairly easily in AD&D, by treating human/dragon as a multiclass combination and giving class-specific XP awards for each).

Systems may be paranoid about xp awards leading to discrepancies in character power; Savage Worlds hands out xp at basically a set rate of 2/session, with good playing being instead rewarded with 'bennies' (luck rerolls). 3.x D&D hands out extra XP to low-level characters to allow them to catch up. This is particularly a concern where xp progresses at a linear rate. AD&D instead has xp requirements that double from level to level so that low-level characters in high level parties will advance faster - if they survive. Palladium gives only minimal power increases from levelling, and has very slow advancement (Palladium like AD&D slows down character advancement for more powerful characters with a worse xp table, but its debateable if this works since level advancement is so slow/benefits limited - a legacy mechanic; its power curve is also low enough that its hard for the GM to make super-NPCs or BBEGs, meaning that they have either massively inflated levels or bunches of ad-hoc perks as well to compete - 'is secretly a dragon' happens pretty often in either the fantasy or Rifts versions).

Superbabes starts characters at 0 xp, with only 1 xp to 2nd level - characters level up at the end of their first session automatically if they did anything, then progress fairly slowly.
German game Midgard reportedly has three sorts of XP awarded for combat (damage output), spells (energy invested), and general skills (successful use) respectively, which are reinvested back into new skills of the same type.

XP may be individually awarded, or XP may be calculated for the group then divided. Original Recon for example does that, even though XP is awarded for things like 'using skills' individually (maybe they learned something by watching, but without GM oversight on expenditure afterwards this could be weird).

XP varies between being given out in very small amounts (like 1-3 session) to huge amounts e.g. 2,000+ for a single monster. (Sharp Swords & Sinister Spells actually rates level in terms of number of adventures, e.g. 2 adventures are needed to go from 1st to 2nd level; Arduin Adventure also does this, much earlier).
Large XP amounts traditional in early D&D were used so that XP = GP could be awarded. Challenges such as monsters may have a fixed XP reward (2E D&D), a subjectively-allocated award (Palladium, 300 for a 'major menace') or 3E uses an awkward table of CR vs. character level to award XP - which is designed so that a set 13.33 encounters is needed to advance in level with the XP per level also advancing evenly (level x 1000 to get to the next level) i.e. there's a calculation required every time XP is awarded, rather than once in class design, although it also sets the relative value of XP so that XP losses from magic item creation or spellcasting are enforced.

XP may be directly spendable to give boosts to rolls or rerolls which may be another factor in determining how XP is scaled, although IMHO this design has some serious problems (cf. 'safety valve' discussion later).
 
Level-based systems where maximum skill value depends on character level create a ceiling to character abilities which controls unbalanced skills i.e. weapon or magic skills very well. In contrast, systems where characters are free to distribute points as they wish are much more min/maxable. Level-based skills systems struggle somewhat with languages (usually these operate outside the skill system, as otherwise a skill level equivalent to "fluency" might not be available to a starting characters!)
Usually in a level-based game, higher level is better. There are a couple of games where a lower number for level/rank represents more ability (Weapons of the Gods, Lejendary Adventures; Generation in Vampire). This unfortunately makes character progression more awkward in terms of calculations (i.e. a minus gives plusses to other factors) and puts an upper limit on ability. Though Weapons of the Gods has one bad guy NPC who is "below first rank" anyway. (The term 'tier' seems to imply this sort of reverse progression, although Numenera uses it anyway with a regular start-at-one-and-work-up level system).
A couple of other level-based systems that work in unusual ways:
*Tunnels & Trolls 7th Ed. generates level from attribute scores connected to their class i.e. a wizard could use their INT or mana/power attribute (/10) to calculate their level. Level affects magical ability and also increases other talents. Characters spend XP to raise attributes directly (current score x 100). Note the system is quite different to earlier versions of T&T, where level gains improved attributes instead.
*Slightly similar in Lords of Creation, XP was spent to add dice to ability scores, with total ability scores determining "Personal Force" which then gave level, number of skill points and powers.
*Elder scrolls (the computer game) has direct skill improvement through training, books and skill use. Each class has a set of skills which must be raised to increase level, putting up stats/HP/mana. Characters can put up other skills (e.g. a wizard putting up fighting and armour skills) but don't get levels for that. In a sense the effect of this parallels the Talislanta system.
*while in most systems there is usually simultaneous 'vertical' (numerical increase) and 'horizontal' increase (number of specific abilities), the Lone Wolf gamebooks were interesting in having increases in Combat Skill or Endurance as a secondary effect of choosing particular abilities.

Systems can have multiple advancement tracks e.g. SenZar separately awards character points (to raise stats or gain special abilities) and XP (putting up level).
The "E6" mod for 3.x D&D caps characters at 6th level, but allows 6th level characters to earn bonus feats with XP, i.e. it radically changes progression system.

Class-based systems that use XP can have very different "advancement tracks" for different characters. Point-based systems are more inherently standardized, though the game may try to work around that by having extra skills etc. that are fairly race or type specific, by having some skills etc. work slightly differently for some characters (perhaps extra bonuses), or by a 'tax' where you need to continue to buy particular feats or edges or whatnot for some concepts, or by purchase limits for some characters (e.g. synthetics in d20 Gamma World can't buy feats relating to 'personality development' after character generation, without first taking a 'Spontaneous Algorithm feat; this also generates costing problems since the same feat effectively costs 1 feat if selected at 1st level, or 2 feats later, while other feats are unaffected). Costs can also be modified for some characters (which I wouldn't really recommend). Unintentional differentials can develop if one character type has a continued need to 'pump up' a particular stat or skill to keep up, when another doesn't.

Negative Advancement: very rarely (at least, aside from ageing) a system is seen where some characters have some sort of negative advancement i.e. gaining experience lowers a characters abilities. (This is different to the case where level counts backward but actually still improves the character; by negative advancement I mean a character actually weakening as they gain XP). A couple of examples of this would be the Reformed Demon RCC in Mystic China for Palladium, which becomes less demonic and physically weaker as they level up, eventually retiring and becoming a new 1st-level human character, or the Risen Martyr in 3E D&D Book of Exalted Deeds, which returns to the heavens (forcible retirement) after reaching 10th level in the class. As noted earlier, Rolemaster has checks for stat improvement that can sometimes result in characters losing stat points instead, but rarely. Basic's divine ascension rules made Level 1 deities worse than Level-36 basic characters in some areas (see 'Divine Ascension post). Palladium also has a "Juicer" class, which had a limited (7-year game time) lifespan; the Savage Worlds' Rifts conversion reportedly replaces the fixed cut-off with a mechanic of "burn", which rolls for decrease each session (and can potentially be spent) until a character reaches 0 and dies.
Somewhat related to this but less extreme, a system can have level 'benchmarks' where a character is still perhaps getting bigger numbers, but not as fast as monsters or other characters do. This is something that was especially noticeable with e.g. 4E D&Ds sample Skill DCs in some places, or monster defenses if Expertise rules were not in use.
A few games have skill atrophy rules for disused skills, e.g. Rolemaster optional rules in one of the Companions IIRC.

Skill Costs: in games where skills are purchased, the skill cost can be based off class skill list (3E D&D, Rolemaster), current rating (with or without adding in attribute bonus e.g. Star Wars), a controlling attribute (Savage Worlds), or skill inherent usefulness (2E D&D, Synnibarr, GURPS). Tri-stat reportedly adjusts costs of skills depending on genre e.g. combat skills will be more expensive in a combat-driven game.

Retraining: as well as adding to skills games sometimes let characters 're-train' existing skills. D&D-4 has built in retraining of powers, with higher level options replacing lower-level options while D&D-3 had it as an optional rule in case of bad choices only, and in the Expanded Psionics supplement has a psionic power that lets a psion spend XP to adjust skills and feats ("Psychic Reformation"). Legend of Anglerre reportedly also has skill readjustment rules. In some cases 'retraining' can make sense if an ability overlaps another ability gained (a character spent a feat to learn battleaxe and then multiclasses to fighter. Likewise a character with Toughness gets several HP from levelling up, or a character with a +1 to hit from Weapon Focus who increases their attack bonus with all weapons when levelling could 'retrain' without this resulting in any values actually dropping).

Training/practice can sometimes be mixed with an XP approach - levelling up can require training. Particular skill use or circumstances can alter level up options e.g. Mutant Epoch allows a character who has attempted unarmed combat during gaining a level to switch their level up bonus (normally rolled randomly) for an extra Brawl skill point. 4E characters who met some FR NPCs (Drizzt, Jarlaxle) could learn specific reward powers taught by them in place of the normal class choices.
Another version of "fiction" around levelling is the contest approach - AD&D 1e in a couple of cases (Assassins, Druids, Monks) required a character advancing to a new level to beat up and replace someone of that level in a duel. The approach probably works badly with its multiclass system in that someone who's multi-classed had a major advantage over a single-class character in a duel (though magic items, ability scores, or other unique features like psionics could also swing the balance).
Another skills-based freerpg, FURPIG, allowed there to be only one 'perfect master' of a skill (100% rating), so someone advancing from 99% had to defeat the current master.

Energy Drain: (A)D&D is almost unique in having it possible for characters to lose levels from "energy drain" e.g. from undead such as vampires, wights or spectres. This is something that works in the older context where all new characters start at 1st level - as a setback less serious than character death - but by 3E D&D is literally a fate worse than death since a dead PC would be replaced by one of the same level. It works better in the context of AD&D since a) replacement PCs would instead generally begin at level 1; b) the XP costs/level double each level so that lower-level characters generated either way can catch up and c) abilities are less level-dependent, and players had opportunity to participate based off player skills rather than character abilities in many parts of the game anyway, making level less important.
D&D energy drain occasionally managed to create characters with greater-than-maximum hit points, due to rolling low when levelling-down than high when regaining a level. [a particularly combo with 'Gifts' from 2nd Ed. Complete Viking, giving a bonus on die rolls of a particular die size, potentially including HP).

This sort of "level loss" can produce weird problems or cases at times.
For example, Gamma World d20 has a "Mastermind" class which at level 10 ["Apotheosis"] lets a character take over bodies to use them as 'nodes' backing up their personality. If a node is killed they lose [total level/# of nodes they have], meaning they would technically lose the apotheosis feature while still having other 'nodes' left.

Ageing: characters may suffer changes related to ageing in addition to earning XP (though some systems, including LEG's Aliens game and Clash's Starcluster/Outremor system, have advancement primarily through ageing e.g. the former has an improvement roll each year, the latter lets characters lose stats but gain skills with age). D&D notably has aging adjustments for each of 1/2, 2/3 and full 'base max' age, where characters lose physical stats and gain mentals; other simpler RPGs omit this and a characters' initial stat array can be assumed to imply their age (e.g. high mental and low physical rolls = an old character). Palladium/WHFR have primarily penalties for ageing (including to Int or equivalent), RQ has random stat losses rolled for each stat and where characters dropping to 0 die (including APP, so  characters can 'ugly to death').

Player Skill again: as well as mechanical increases, players in any system can improve in terms of using their characters' abilities better. This can be more pronounced in rules-lite games where the actual scope of abilities is unclear (e.g. Amber Diceless specifically makes mention of it - noting that characters with Shapeshift from different campaigns might meet and be surprised at what the other can do with the same power). Some player skill improvement may be 'in world' learning, some may instead be a matter of better understanding pure game rules (e.g. learning to take-10 more often).

Note: see also here for overview on character advancement-related topics.

Note that character advancement can sometimes also trigger or require improvements in other characters e.g. a replacement character or another player character. A replacement might be created at the same XP total/level, gain a portion of XP/levels, or gain an ad hoc bonus to base skill. New characters added to a party might also be made equivalent. Interesting systems here include Amazing Engine's "player core" and Dark Sun's character tree (mentioned briefly in attribute generation).

recent edits: 24/12/2015 - picaro thread link

Advancement note: with XP type systems, skills might increase immediately or require training as well. FantasyCraft has a feat ["I Can Swim"] which lets characters adapt on the fly by not having to spend all their skill ranks immediately after levelling. This is somewhat workable as a feat just since a character gets a new shipment of skill points with each level advancement, although its utility is limited compared to D20 since FC characters can't buy up cross-class skills at all.

Advancement systems can have implications in terms of character background design. e.g. a level 1 character is sometimes assumed to be fresh off the farm, with the assumption they haven't fought any monsters or done very much; this is worse if identifying traits like 'is a pirate' can't be gained until level 6 or 7. [3.x prestige classes]. I usually like to not take that too far, and assume that mechanics aside, a 1st level character can still have done a few things. Note that 3.x is also weird in that it has a very high power curve where some PrCs are limited to high level, despite play already having changed enough so as to make many of these less useful; these may really work well only for NPCs, or where a party ranges around in level.

Most of the character building process - attributes, skills/classes, etc. - seems to have relatively little effect on choice of core mechanic  (d100, dice pool, etc), and vice versa - games are fairly similar and follow similar ranges of choices. In very rare cases, an unusual core mechanic might force a strange adaptation e.g. Marvel Super Heroes' universal table system has rolls based off a single attribute only (not stat+skill, etc), and hence has a Fighting attribute; DC Heroes' system needs Acting, Effect and Resistance numbers from attributes and so has lots of similar-sounding attributes (e.g. Mind which gives a target number to avoid mental damage, plus WILL to absorb damage if it occurs).

With regard to skill and level systems, note that:

*level-based systems are not usually dice pool games - in part because the dice pool games appeared in the 90s when levels were going out of vogue, but also perhaps because dice pools don't scale to large numbers (level would have to set dice pool directly) and so can't as easily account for skilled/unskilled as well as level and attribute. Dice pool games usually don't have inflationary HPs either, removing another reason to have levels; soaking is easy to implement by counting successes so most do that.
I know of no true level-based dice pool games (aside from T&T with its multidie-additive combat system), although Storyteller games do have a level-equivalent primary stat such as Generation for vampires, Arete for mages, Rank for werewolves, Quantum for novas (Aberrant RPG), etc. Also interesting is 'Over the Edge' - here characters have an experience pool which acts as a reserve of points that can be spent to get bonus dice, rather than adding to every roll.

D10-based or systems using changing-die-type are also not usually level-based due to high granularity- the exception being Earthdawn (step-dice with rank determining sets of dice rolled) and DCC (which uses d14 -d30 as well as level bonuses). d10 provides almost enough granularity for level bonuses, except that skill bonuses from level would tend to be outweighed by attribute mods unless attribute scale is limited to a very small range.

The concepts of 'levels' and 'classes' usually go together - partly perhaps due to D&D traditions, though it has also been argued that skill-based systems do not generate "level appropriate" results, and in general games without levels rarely have 'classes' - they are more likely to use 'archetypes' or free skill selection. Most 'derived attribute' games are level-based too, so classless games are more likely to be skill-driven. Fuzion is sort-of an exception in having lots of 'derived' attributes that are averages, although its non-level-based and uses d10s (if not using the 3d6 variant).

Other notes:
Point-buy systems get increasingly unwieldy as stat scale goes up to larger numbers (for use with 3d6, d20 or d100), although to be honest this has never really stopped anyone.

Advantages/disadvantages tend to have more of an effect in dice pool games or games that are more granular - comparatively additive systems give out smaller bonuses (Compare White Wolf advantages or Savage Worlds' Edges against lots of the d20 system feats, which frequently give out very small bonuses). Racial stat bonuses are also more pronounced (e.g. Shadowrun 1E).

Usage-based improvement isn't usually seen for highly granular stats (step-die, d10, or dice pool) although a system could award 'xp' towards an increase from successful use. Random improvements are most common with d100 systems.

Highly 'abstract' statistics (D&Ds six attribute system) perhaps don't work as well with detailed skill systems -the skill modifiers limit how the player can interpret a low/high attribute (e.g. 2E might let a character explain their low DEX as due to being half-blind and so a lousy shot, but DEX in 3E doesn't modify Spot by default).

One-Value vs. Two-Value Systems
A core mechanic has a 'base chance' of success which could depend on a single number, or may be calculated by adding two or more numbers together. Table-lookup games for instance ( Marvel Super Heroes, DC Heroes ) typically use one value. FUDGE uses one value corresponding to a descriptor (Good, Great, etc.).  Likewise, roll-under games often use one value - d100 games are an exception in that while there are d100-based games that use just attribute checks (Maelstrom, Amazing Engine), a percentage is more often calculated in a more complex fashion (e.g. Runequest). GURPS (3d6) would be another exception as it uses a table to factor in # skill points as well as base stat, though stat still has a high weight). One-value games generally correspond to the 'attribute-driven' model [FUDGE being the exception] whereas two values (see next) relates to stat+skill.

One Roll Engine or Storyteller (or Toon) however inherently use two values (Stat+Skill) -in these games limiting to one value is effectively a major penalty. (Many other games can accomodate either one or two values - here having two on a check is virtually mandatory).
Toon uses 1d6 to determine stat and 2d6 roll under for skill checks, which gives very low chance of a raw stat roll succeeding with the same mechanic - less than half that of a skill check due to the probability curve of a 2 die roll. One Roll Engine chance of success likewise increases dramatically as dice pool increases, though as stat and skill are both 1-5 they could've used [stat+stat] rather than inventing new skills (Cortex is similar but sometimes uses stat+stat (cf. Serenity pg 142), although it also has a number of skills that might be classed as filler, such as Athletics and Willpower).
This therefore generates a need for extra values (or very broadly defined values) to ensure all tasks have a reasonable chance of success, or will have 'holes' where chance of success is too low...raw Str checks, perception checks for taste/touch, etc. Skills can be forcibly created to fill gaps if rolls must be [stat+skill] and never stat alone.  
Fuzion also has a number of skills which fill this niche; these usually default to 2 though with it being argued that scores should be higher for more competent characters e.g. =attribute (which would only work due to stat/skill being on the same scale).
Skills are also sometimes invented for the opposite reason e.g. if a raw stat check is much more likely to succeed than a skill check - this may be the case with BRP, where stat roll is usually [stat x 5% or around 50%] whereas skill is a small base score % + stat modifier)(often under 20%).
From a character-building perspective, two-value systems may be slightly harder to build effective characters for, since making a PC good at something involves putting up both values, possibly in conjunction with knowing the relative importance of each (weight on checks, etc). Poorly costed build options could easily have a low attribute invalidate a large-ish skill point investment or vice versa.
Going from using 'one value' to 'two value' is if anything more difficult than the other way around. Sometimes a game will use averages (for instance, Tunnels and Trolls in some solo adventures averages two scores); sometimes a game will apply a small modifier from one stat to another that doesn't totally break the scale, sometimes a game will drop the core mechanic and go to a different mechanic (DC Heroes initiative of d10+sum of the three 'acting' attributes, rather than the usual table).

[Games with one-value can still add lots of small modifiers - e.g. Marvel Super Heroes' 'column shifts'.  There can also be games where instead of one main value, you are expected to reference and add up a bunch of bonuses  (e.g. say Cortex+ with its various dice), which you could call an n-value system. How well a character does might be harder to predict in advance since its a question of how many bonuses can be brought to bear, rather than how big one number is, though Cortex has safeguards to make sure some dice are always applicable. This sort of approach probably requires defining lots of specific aspects for characters; the idea might be to get around people breaking the system with extra modifiers by making the base value low so you need all the extra modifiers. Simple additive systems trying to use this can get fairly broken - maybe 3E D&D armour class at higher levels, where PCs had to deliberately stack various AC bonus types to reach the right AC]

Probability Curves & Skill Breadth
Core mechanic can also influence what breadth of skills makes sense for a system. Both D&D 4E and Pathfinder, for example, have gradually adopted broader skills in conjunction with their D20 mechanic.
With very narrowly-defined skills, randomness is less desireable, while for broader skills, how much the character knows about the exact sub-topic is a variable that's part of the random roll; in these games the random roll is skipped when/if likely to give inappropriate results. In another words, if you're playing GURPS then having a priest of Pelor make a Knowledge: Pelor roll would give probably reasonable results (since this uses 3d6), while for D&D 3E they could botch this and know nothing about their god, so it makes more sense to have a Knowledge: Religion skill and have the priest not roll for Pelor-related tasks.

Core mechanics with limited scaling potentially limit multiclassing (if benefits from levelling are low, then gaining a new package of abilities instead is more unbalanced).

High-order Considerations
Systems could be broken into 'Concept-first' or 'mechanics first' games with regard to character generation - in one you start with mechanics (an optimal feat chain, multiclass combo, etc) and interpret a character around those, in the other you start with a concept and find mechanics that fit. Which applies depends on how many mechanical 'moving parts' a system has, how balanced options are and how re-skinnable options are. Factors like how fuzzy attributes are, how broad classes are, and whether or not there is a feat system, are all contributors.

Systems also vary in how inter-connected various character options are. A highly detailed system can have rules where e.g. size modifies Strength directly, whereas another game e.g. Storyteller lets a character buy the Huge merit and Strength separately, and the onus is on the player to create a final character sheet that makes sense; this gives the player leeway to adjust for various factors, and to consider connections that might not have been accounted for by the designer, but also can be a temptation for players to build for optimum character power instead of matching a concept. Another example is the split between free choice skills systems and those with classes that are tightly defined; a character in one picks a class e.g. doctor and gets a number of skills with not much choice in the matter, while in the other a character just chooses the skills that match the concept (so they can fine-tune specific concepts, but could also overlook something important).
Detailed rules for modelling interactions between various bits on the character sheet can be very specific and fiddly, and can't account for everything, but maybe should be used where details are something that should be niche protected or relatively rare.

When designing, there are often multiple methods that can be quite different yet yield similar results. That is, the systems are structurally different, but functionally the same. For example, an emergent final result of "warriors typically have higher physical scores such as in Strength" could occur in a game where fighters get a bonus to Strength as a class feature, or as a result of some classes costing fewer character points to buy and the same points being used to buy attributes, or by adding an extra 'Power' attribute that's necessary for wizards but a "dump stat" for warriors. These options could differ in some specific edge cases, for example a bonus to Strength as a class feature will (barring extra rules) give multiclass characters more Str, the second option less Strength (having spent points on buying two classes) and a middling result in the third case. Similar results in play can also occur from different amounts of starting complexity, e.g. based on ad-hoc adjudication as opposed to numbers of formalized rules.
(Note: Here, we're speaking of comparing systems. Potentially there might also be multiple ways of generating/representing something within the same system [like HERO characters sometimes able to build the same thing more cheaply with Power X and some disadvantages, than Power Y), but this is probably a flaw since it can lead to 'right' and 'wrong' ways to do it).

Systems sometimes have specific 'order of operations' where you have to do say stats, then class, then skills - an example is AD&D 2E where you need particular stats to qualify for a particular race, then race affects class, then classes buy skills for different costs (Skills and Powers was particularly fiddly that way in that points could be moved across categories that way but only to later 'steps', with advantages coming weirdly late in the process after race and class purchase and so being easier to buy if you have a class/race with a larger Character Point budget). In another game skills, race etc. may form different 'partitions' but in no particular order - say Storyteller where a character buys skills the same regardless of what their vampire clan or mage tradition or werewolf auspice is, you could start anywhere really.
Order of operations (e.g. stats > race > class > advantages) tends to be more annoying if a character makes a decision and then finds it invalidates later choices, needing redoing - for instance, if stats are bought and then the player discovers they can't qualify for a feat or ability ('charm', etc.).

I think optimally choices should also be arranged such that they are a) independent (avoiding trap options where one choice early on, locks you into another particular choice later - classes and essential attributes for them would be an example, though one that's hard to build out short of using archetypes and b) not degenerate - for instance having a choice between A/B/C and then between D/E/and A again, such as if a feat duplicates a class feature. Probably better to narrow this to one decision point. The peril of duplication is that two different characters might have the ability, but with different trade-offs to get it leaving one in a worse position.
(in the past I did ignore that for one design, where a character could pick up 'Luck' later in character generation as a trait, or earlier as a Talent [Talent meaning not a skill but an aptitude for particular skills] which gives points in the trait: the main difference here was that Talents occur earlier in the order of operations, in this case with a random disadvantage-determining step in between: Lucky the talent like other talents allowed a character to reroll some particular results on the random disadvantage roll which Lucky the trait normally couldn't, simply because buying Luck is normally the last step).

Systems in some case might design a number that's an 'input' initially, and in other cases derive it as an 'output' at the end. Mainly depending on what you want to use each for; for instance 'Education' might be an attribute that at the start gives a number of bonus skill programs, or it might be a derived stat [/thing] calculated from what skills a character has bought or the bonus of them, with a bonus for other feats/advantages, then used to modify the reaction rolls of scholars or a character's default %knowledge checks.

Number control (abbreviated from later)
When trying to balance or control increase in numbers, a number can be set in various ways, essentially with a continuum of a variation that goes like this:
*fixed for all characters
*fixed, but with possible modification for a rare Advantage if bought/rolled.
*archetype-based numbers (meaning stats can be controlled in an archetype-specific fashion - if a character type gets massive extra value from CON e.g. some powers are based off it, it can be 'costed up').
*An attribute, which can be rolled or bought up at cost in other stats.
*A number which automatically scales up based off 'level'.
*A number which can be purchased upward with variable expenditure of points.
[this is rough, since there's actually two sorts of scaling - horizontal between characters and vertical with 'level' or experience - and something can scale by both or neither]

Niche Protection
Its debateable as to whether 'niche protection' is desireable in the first place, but something else to note is that games where roll bonuses are highly divergent between characters tend to produce 'niche protection' automatically (to be competent at something requires a larger investment). Generally, high variability in a roll cuts into divergence (anyone can roll high and get lucky). Basically then D&D can be thought of as creating niche protection 'artificially' where a skill-based system often does it innately.

Synergy
Options intrinsically work or don't work with other options. Competition for varying resources is usual, for example abilities that are based on different attributes; action-economy where you can either swing a sword or cast a spell, or so on. Abilities can directly synergize, be redesigned to synergize more or less. It can be possible to promote a synergy directly between A and B, or create an option C that helps both. Implementation of a specific ability can often be fine-tuned so as to promote synergy or not - compare 'elves get +2 to Spot' (3E D&D) and 'elves get Perception skill for free (5E D&D) - both making elves more perceptive on average.

Edit notes: niche protection (*), redundant options with system (*), different ways to control numbers (*), more on order of operations(*), n-value systems (*)

General intro on Resolution Mechanics
Every mechanic is going to have its pros and cons.
Any single-die mechanic will have very swingy results (including 'changing-die-type' systems, these being if anything more variable than die+mods systems).
(Often in practice, tasks may incorporate multiple rolls or allow a certain number of rerolls - 4Es 'skill challenges' for d20 being an example - going the other way to make 3d6 more swingy is a harder task!)
Any multi-die mechanic will be annoying to roll for lots of targets/foes at once (...though they may sometimes at least generate a couple of results simultaneously, as with One Roll Engine combat, Warhammer 1E/2E reverse-d100 hit location, or Heroic Golden Turbulence colour results; see later). It will also be less transparent with respect to probabilities (though it is sometimes argued that less transparency is desireable, to keep players immersed as opposed to calculating probabilities; quite often even linear dice rolls are made against secret target numbers to obscure probability, e.g. D&D hit rolls).
Take-highest systems are both swingy and clunky.
Diceless or card-based have their own issues, like being overly deterministic or needing shuffling.

An additive system means more math.
A roll-under system will be 'nonintuitive' (lower results are better, unless its a 'blackjack' system), gives extra subtractions instead of additions as soon as modifiers start applying, and are generally more work for opposed rolls (e.g. initiative).

There are also trade-offs between lower and higher granularities (size of dice). The finer your resolution (to handle detailed advancement such as level advancement, or multiple modifiers) the more the math multiplies (for e.g. adding bonuses, or converting rolls back to success levels at the end). At lower resolutions odds of critical hits/fumbles occurring, or alternately of needing an extra roll to 'confirm' your success or fumble, may get too high.

Choosing a core mechanic
Usually 'core mechanic' is something chosen based primarily by designer preference, rather than any objective reason.
Some interesting cases of why particular mechanics were chosen include:
-Dogs in the Vineyard; its dice pool system is built around specifically having social rolls that fail and turn into gun battles.  
- Savage Worlds the core mechanic (step die) was chosen just so mass combats run more quickly.
-DC Heroes uses tables that cross-reference scores to get results, as part of a universal result system designed for comparing logarithmically increasing attributes. It works great for figuring out how far a STR 15 character can throw a Size 5 object, for instance.
-One Roll Engine has its height/width counting which is used on every check, mainly there to do hit location as part of the attack roll (a bit gimmicky).

A note on distribution curves
A single die roll generates a flat curve, whereas a multiple dice roll generates a bell curve (e.g. 3d6). The single roll has the advantage of having odds that are easily calculated and understood (i.e. on d20, a +1 always increases a characters chance of success by +5%); the bell-curve on the other hand generates results that are more predictable - a dice roll will typically be an average result (i.e. 9-12 on 3d6 is much more likely than getting a 3 or an 18).
While the d100 generates a flat/predictable linear curve, the fine detail allows for considerable control, e.g with 'special successes' and the like set to a fixed (small) percentage (1/10th of regular skill, for instance). Or tables can be used. Such methods can transform the linear roll indirectly to get 'diminishing returns', i.e. a non-flat result pattern.
Some interesting discussion on the effects of how modifiers affect probabilities in true bell-curve systems (like GURPS) is here: http://forums.sjgames.com/showpost.php?p=1490660&postcount=7

Opposed rolls in additive systems where highest wins basically converts a linear roll into a curved distribution (e.g. d10-d10 = same spread as 2d10). Opposed rolls in roll-under systems where a number is a flat succeed/fail (e.g. Runequest for the most part) are probably more complicated.

See below for more on specific implementations.
EDIT NOTE: See also post #42 (combat manuevers and core mechanics) for further on combat system design specifically.

D100 systems.
A d100 system is finely detailed, meaning there are no break points/dead spots for attributes – every point is important, limiting min/maxing. Using a percentage can make it easier to consider multiple factors (such as several attributes modifying skills e.g. HarnMaster), or different weightings i.e. one stat could give +2% per point while another gives +1%. The very exact percentages may be useful for critical hits or fumbles, etc.

It is slightly slower for multiple rolls (as dice must be paired up as 10s/1s). (Potentially, a d100 additive system can be sped up slightly by rolling d10+modifiers, with a fail-by-1 triggering a reroll against the ones place.)
d100s may be used as roll-under or additively; see relevant sections for further notes...(post 12 directly below & post 14).
 
With values being expressed as a percentage, players can immediately see their base odds of success in a way they are not able to with, say, dice pools, and new players can usually grasp immediately that say, 85% library use is good. (Though not all systems using a d100 are necessarily this transparent - tables like in Marvel Super Heroes can make the %s less apparent).
% systems tend to be upward bounded at 100% (i.e. making it difficult to exceed this); so that penalties having more serious effects on higher skill characters i.e. a -20% drops a character with 95% to 75%, or a 1-in-20 failure rate to 1-in-4, while the character with 60% succeeding just over half the time drops to 40% or just under half the time. Bell curve systems are sometimes recommended to get around this sort of problem.

d100 systems occasionally generate additional information using the '1s place' of the d100.
-HarnMaster treats rolls ending in 5 or 0 as critical successes or critical failures, to assess these quickly (if non-intuitively)
-Amazing Engine treats the 1s place as the quality of the result; for weapons if the ones is less than weapon "lethality" it deals "body point" damage rather than "stamina point" damage.
-Warhammer 1E/2E inverts the attack roll (i.e. 39 would become a 93) and uses this as the hit location roll.
-early versions of HARP (discussed on Rasyr's blog wizlair) used d100 + mods to attack, then added together both dice to get a 2-20 number for hit location.
-Top Secret/SI uses the 1s place on attack rolls to determine hit location, and the 10s place indicates damage for melee attacks (higher is better i.e. blackjack success); firearms instead deal random damage (thanks RobM for this note).
-a number of games use 'doubles' on the roll to indicate a special result (Unknown Armies; Rolemaster weapon breakage).
-a few games (e.g. WHFR 'clone' Zweihander) can allow a switching of the 1s and 10s dice as a special ability - effectively an instant reroll (though with no extra chance of doubles). Unknown Armies has skills letting this happen provided the switched result is under the skill, so higher skills can increase chance of a flip (& multiple skills feed into one skill check).
-with a blackjack system where higher is better, flipping d10s/1s place could increase chance of success but at a cost in degree of success (Effect).
-If the 1s die is used to determine "effect" [e.g. HarnMaster, Amazing Engine type mechanics], then switching the 10s/1s becomes more frequently possible when the task succes percentage is high, meaning that there's actually an increase in effect from high skill value instead of it being effectively independent. For instance with Harnmaster's '5s are crits' a roll of [6,5] would be a success given 58% skill ('roll' of 56%) but would be a crit at 65%+ skill ('roll' of 65).
-PlanetAlgol on his blog has suggested having the player roll the 10s dice and the GM the 1s dice to preserve uncertainty (though this works only sporadically)

d100 systems sometimes have attributes that are directly set up to be used as percentages rather than being skill-based (Amazing Engine, Warhammer 2nd ed), or they may have ability scores rated on a lower scale such as the 3-18 of Runequest/Harnmaster. These last tend to use a formula of [attribute score x a difficulty multiplier] to determine a base percentage chance for attribute checks; HarnMaster sometimes just uses a 3d6 roll for ability checks instead.
DragonQuest (at least the 2nd edition of it) has a statistic scale that's slightly higher (going up to 25 for a normal character and includes fractional multipliers e.g. x2.5. To prevent characters going past 100% too readily, an additional rule gives a minimum chance of failure equal to (30-stat) i.e. a character with a 25 Strength would still have a minimum 5% chance of failure on a Strength check, meaning a roll of 96-100 fails even though the character would have a base 100% chance of success at e.g. [Str x 4].
 
d100 systems tend toward expansive skill systems with lots of tightly defined skills. My theory here is that either a) these systems tend to be built by designers focussed on gritty realism down to the 1% level, or b) these games need to export stuff that would ordinarily involve an ability check off to the skill system (like Listen in BRP) since leaving it as an ability check would make the % much higher than other skills.
This and the scale of skills to start with can mean juggling fairly large point budgets during chargen, e.g. Call of Cthulhu needing to divide 300+ points between various skills.
 
Another interesting rule for d100 is from J Arcane's Drums of War: this rolls 2d10 additively, but uses the same numbers to generate a percentage for the purpose of determining if a critical occurs (i.e. 9 + 7 would be a 16 to hit, or 97 for criticalling, compared against the characters % chance of critical).

While a %-based system's fine resolution means lots of small modifiers should be applied exactly (+1% circumstances like slight wind on your archery check) this is a bit fiddlier with roll-under than with an additive system. Modifier heavy games get slightly clunky with d100 (e.g. Deathwatch).

[Edit] Final Note: while it seems I'm against 'swinginess', it can be a legitimate design objective in and of itself - to generate a range of game outcomes - although it can be at odds with either minimizing lethality or generating balance. A case of a game deliberately designed to produce weird and swingy results is the DCC rpg - a good sampler article is here:
https://muleabides.wordpress.com/2010/12/13/glorious-swinginess-results-from-the-dcc-rpgcastle-zagyg-experiment-part-1/

Additive systems are excellent for opposed tasks and provide intuitive results; higher numbers are better. Multiple dice can be rolled at once more easily than with dice pools (to allow the GM to, for example, roll several attacks at a PC simultaneously, or several saving rolls) but perhaps slightly less readily than roll-under systems. Level of achievement is more prone to inflation (usually, amount by which a roll succeeds gives effects of success). Probabilities are slightly less transparent than d100-roll-under but better than dice pool systems, at least if using a single die.
Some additive systems suffer from out-of-control bonus stacking.
Additive systems may be slightly better than most other types of mechanic (e.g step die, dice pool) for letting players roll without telling then for what.

Elaborations:
Additive systems can provide success levels as a proportion of success chance by having a natural die roll that is ‘confirmed’ by a successful check. For example a longsword in 3E D&D used by someone with a 60% chance to hit (e.g. +4 to hit vs. AC 13), has a 6% chance of being a critical; a natural roll of 19-20 triggers a possible crit (10% chance ) times a 60% chance of hitting (confirmation roll) = 6%.

Combination with other setups: 5E D&D uses d20+modifiers (attribute, proficiency bonus), and also frequently has a rolled-bonus for some class features e.g. sorcery points (d4), bardic inspiration die (d6 base) or fighter 'superiority dice' (d8 base)  - with dice increasing sizes at various levels. Increases are fairly erratic, and whether a class bonus is a flat + or a die is fairly ad hoc. In some respects reminiscent of Alternity (a roll-under system) but less structured. (Also compare 'action dice' in FantasyCraft - noted in safety valves). The main benefit of the rolled bonus here is that it can often [depending on the specific power] be declared after a roll is failed, still with some doubt as to whether the boost works...e.g. you could fail by 2, then roll a 1 on +d6 and still fail. This wasn't a feature of Alternity, where the step die was rolled simultaneous with the main 'control' die.

Variability: How variable the outcome of an opposed roll is, in an additive system, can be reduced by allowing all participants multiple rolls, taking the highest result (or the middle result). This skews up the average toward the maximum; the likelihood of the character with the highest bonus winning increases. This could be used for tasks where results are more cut-and-dried than normal (an opposed Strength check for an arm-wrestling contest, for instance).
This same method can be used in roll-under systems that use margin of success.

Whereas in roll-under systems the total chance can be multiplied (e.g. halved), an additive system instead makes it possible to multiply either the bonus to the roll, or the base target number (this is very rarely seen). Halving the target number is equivalent to averaging the percentage chance of success with 100% for a character with bonus +0. Halving the bonus on a DC 20 roll (for d20) is equivalent to halving success chance.
With a linear die + modifier, adjustments to chance of success have a fairly coarse effect (its easy for a modifier to push a success chance beyond 100%, and difficult to set exact %s as desired) : Timelords had a modifier table which cross-referenced bonus and difficulty to give a proportional reduction, so that highly skilled characters didn't automatically succeed at tasks, though similar effects are easier to implement with bell-curve systems (die pools or 3d6) or roll-under.

Modifiers: Perhaps as a general rule the upper end of bonuses should be about as large as the die range (i.e. +20 for d20) so that the upper end is an automatic success - less than that gives everyone a chance of failure on all tasks. This does depend on the DCs set, and means that at the high end d20 systems will need double-digit addition. Actually, expanding range of modifiers beyond [1x die range] may be desired to guarantee predetermined outcomes for opposed roll contests as well (+full die range  is needed to guarantee outperformance, between two benchmarks rather than entire top/bottom of scale, at least if assuming that the comparison is based off the margin of success and not some other effect system).

Here I include systems such as West End Games Star Wars and (for combat) Tunnels and Trolls. A pool of dice are rolled and the total calculated. A problem with this system is that a character’s result varies enormously depending on # of dice in a characters pool – making some tasks very easy or impossible for some characters. To ameliorate this, WEG Star Wars uses a safety value in that characters can spend Character Points/Force Points to add extra dice when needed; Star Wars also adds a “wild die”; one of the dice has a different colour, on this dice 6s are added and re-rolled, while a 1 takes away the highest roll and may result in complication/fumble. deadEarth (also multid6 additive) characters with the 'Unlucky' mutation roll a wild die with a 1 taking away the highest roll, then rerolling with a second 1 meaning 'something terrible happens'.
Theoretically, a 'roll again on 6' could be applied to every die (not seen as a core mechanic in any system I know of, though 2E AD&D's Masque of the Red Death has firearm damage rated 1D6 [Derringer] to 3D6 [shotgun], +/- modifiers, and all 6s add and roll over).
Some T&T characters may have powers that let them 'roll doubles' in combat again (Berserkers).
 
Star Wars/D6 system also added more granularity to its dice pools by having intermediate levels of +1 and +2 (i.e. attributes went d6,d6+1,d6+2,2d,2d+1,2d+2, 3d, etc...(similar to GURPS damage values by STR except that these also had a -1 step i.e. d6,d6+1,d6+2, 2d-1, 2d).
 
Legend of the Five Rings (and I think 7th Sea?) lets characters roll dice equal to [attribute+skill] and keep dice equal to [attribute].
 
EABA rolls, and counts highest 3 in the dice pool. # dice equal stat/3, with remainder added (i.e. 11= 3dice+2, 12=4 dice) (before difficulty adjustments). For some target numbers a +2 may be better than having the [next largest] number of dice, although characters can opt to trade a die for a (non-stacking) +2. Some talents can let a character take 4 dice.  http://www.therpgsite.com/showthread.php?t=31552&page=2

The Framewerk system (Cthulhutech) has a more complex system where a character rolls several dice and may take the highest single die or a sum of a couple of dice, depending. A criticism of the system may be be that which the system is almost as complex as say Cortex+, it does not do anything a [d10+modifiers] system wouldn't do, except for making the probabilities weird and nontransparent - compared to Cortex's (e.g. Marvel Heroic's) various applications built into the system which assign effect, generate complications, recharge pools and so forth.
A detailed breakdown of the Framewerk system's probability issues can be found here:
https://web.archive.org/web/20100611041917/http://taharqa.org/?p=249

Over The Edge uses an additive d6 dice pool, with character advantages potentially adding bonus dice or subtracting dice. A penalty functions by adding a die, but forcing the highest roll to be discarded, so buffering the results from changing too much. It has an (optional) rule whereby any 6s rolled may indicate a 'partial success' despite an opponent having a higher total - the rulebook example being an exorcism that succeeds by trapping the spirit, but gives the ghost a slowly increasing hold on the character. Note: for reference, Over The Edge is (partially) cloned as the basis for free Thundarr the barbarian rpg Under the Broken Moon.
Similarly, 'Don't Look Back' (reviewed in Dragon #220) has a system where a pool of d6s is rolled (3 + modifiers) and 3 are always taken, either highest if positive modifier or lowest with negative modifiers, giving a success rating of 3-18. 11+ translates to success, with weapon damage being (hit roll-10) i.e. base 1-8, times weapon damage multiplier.

Early versions of Imperfekt Games' Invulnerable superhero RPG (the 'Year One' edition) apparently used a multidie d6 system which included rolls of 'hyper-dice' which were [d6x10] alongside the normal d6s, and difficulty ratings going up to 600+. (The equivalent of this for a dice pool system might be e.g. Aberrant, where mega-attributes would get multiple successes per die hitting the target number). The modern version of Invulnerable (the 'Vigilante edition') is instead a combined additive/multidie additive system, where a normal roll is 3d6+stat+skill but with 'hyper-attributes' now adding extra d6s (the rationale given for having hyper-attributes be rolled being that it gives normal characters a slim chance of success). (A setup where powers, including 'hyper attributes' get extra dice like this could be a way to build a fundamental failure rate into powers, i.e. 1 = doesn't work, as a part of the core mechanic/a roll that would be made anyway, though perhaps with the dice roll perhaps still made even on normally automatic actions).
The lost free-rpg ROAR, related to T&T, was (IIRC) similarly a combined additive/dice pool system, with a normal roll being 2d6+attribute but with bonus dice sometimes given out; the two base dice were I think a different colour as doubles on these could roll up and/or a low roll could fumble.

Storygame My Life With Master uses a d4 dice pool, where any rolls of 4 are discarded before dice are added, possibly to give a fairly narrow range of results possible. This gives the same average as d4-1 per die. (A similar mechanism is seen in JAGS which randomizes 0-20 for roll under using 4d6, 6s count as 0, to avoid all rolls being 4d6-4).

A system I saw distribution charted for with the "Troll" dice roller, but not actually as part of any system I've seen, was "4d6, eliminate duplicates and add" -i.e. only different dice rolls are included. (Difficult to say how much more dice actually add to result- maybe best with a fixed-size pool).

A feature of multidie additive systems is that ratings can be used as either a # of dice (the default), or instead adding a flat bonus equal to rating (much lower weighting) - e.g. rating 3 could be rolled as 3d6 or just give a +3. This could be used to represent skill synergies, aid another, etc. This idea shows up in Infinite Power (e.g. the 'dead aim' talent adds Reflexes score, normally a # d8s added together, as a fixed bonus to ranged attack damage).

John Wick's game "Thirty" is a multidie d6 system which is somewhat like White Wolf in that it uses 'Trait' (2-5) + Skill (0-5) for number of d6s rolled. While simple, a lot of dice and results are fairly unrandom (unfair particularly in combat). An interesting rule is that it has "Backgrounds"; a background adds an extra differently-coloured die if it helps, and also any dice that roll less than the background die are raised to that (so if its a 6, all your dice now count as 6s; making the 'swordsman' background deadliest in combat for instance). So far so good, except backgrounds can also be negative, with the extra dice rolled and that setting a cap on other rolls. (With low dice pools however the extra d6, if high, can be higher than if the negative wasn't applied). Potentially making the background die negative could work somewhat if not added the same as if it was positive, and perhaps the background die could be less than d6, say +d4.

Combining Multidie Additive and other systems e.g. Count Successes, match-counting:
Later editions of T&T (7th ed.) include a success counting mechanism as part of combat as well (where 6s count as points of automatic ‘spite damage’, and may trigger monster special attacks e.g. enough 6s from the Medusa turns one PC to stone). This scales up results with increase in attacker ability, but is unaffected by defender ability. The main issue with this is that T&T sometimes uses very large numbers of dice (e.g. the largest monster in the Corgi solo adventures, the Shoggoth, had MR 1000 = 101 dice) and counting the 6s gets a bit wonky when trying to replace e.g. 20d6 with [2d6x10]. Although perhaps in cases like this, Spite could somehow be rolled separately, e.g. 6 dice could also give a roll of 1d6 for # spite 'successes'.
(My own house rules for T&T, years ago, included a star wars-type "wild die" which had a fumble on a 1 and automatic hit on a 6. As T&T uses 2d6 (doubles roll up) for most rolls outside combat, a later idea recently would be to roll the wild die but on a "1", roll a second die and treat this as a Saving roll, i.e. the other rolling being a "2" would be a fail [double-1 would roll up]).

Free rpg Twisting Tunnels ( http://beardedbaby.net/twisting-tunnels-rpg/   )is slightly similar but uses a system combining additive dice and match counting: a roll is made with multiple d6 with either the highest single roll or the sum of all matching rolls giving the total (and compared vs. opponents total). The effect ("Impact Dice") is equal to the number of matching dice, and is often used as a modifier to a follow-up roll, either as a boost e.g. a bonus to attack following from a good Stealth roll, or a penalty e.g. netting an opponent would give the impact dice as a subtraction on rolls until they untangle themselves. Compare also One Roll Engine (post #20 below).
(The Twisting Tunnels system could be interesting with an add-on for giving ORE style hit location: since it uses 'sum' rather than 'count' to give total to-hit, it'd be possible to make head shots more difficult by having these be a lower number i.e. if '2s' are needed to hit the head, it becomes much harder to land a head shot against an opponent, if they can build a defense total from any matching numbers in their roll- a single 6 would block three 2's from an attacker.).

kosmos1214 proposed a system (see page 26 of this thread) where dice are added up to reach a target number, with extra dice remaining counted instead of adding these together to determine effect (e.g. number of wounds). Elaborations considered later included mixing dice types for more variability, or using the highest dice to hit the target number first vs. using them left-to-right.

Final Notes
Overall, multi-die addition systems are good at providing simple linear return (e.g. 1d6 damage per caster level) directly, or less than linear growth if slowed down first by e.g. escalating improvement point costs or the like. This is perhaps more likely to be desirable in rolls for 'how much' (effect) rather than 'do I succeed' (the question a core mechanic has to address primarily). The mechanic does combine well with extra match- or success-counting gimmicks as noted above, but in its basic form doesn't really have much to recommend it mechanically (...unless you want to roll lots of dice) since its non-transparent, requires a moderate amount of addition, scales badly and doesn't give any additional features beyond what most other options do. Multidie tends to happen fairly often in some of the oldest systems, from before polyhedral dice were as generally available.

In these games, a die roll is made under a character's statistic for them to succeed. This is mathematically almost equivalent to using addition (except on opposed checks where either side may fail their roll completely, making a margin of success comparison unnecessary; see post #117). It may be arguably less intuitive (lower numbers are better).
Roll-under may be slightly easier on the players mathematically than an additive system (the player is comparing numbers rather than adding them), particular where rolling multiple checks at once, although any benefit is lost as soon as margin of successes have to be calculated - something players may start to do automatically just in case. Opposed rolls tend to be particularly awkward in roll-under systems; the default mechanism being that both opponents have to calculate how far they rolled under their success number (e.g. GURPS), but this gets annoying for some players, particularly where d100s get involved.
Calculating margin of success with roll-under is the same as taking the stat and subtracting the die roll.
 
Roll-under systems are excellent for applying proportional modifications to chances of success - creating tasks with 1/2 normal chance of success, for instance (though low range roll under, like Lamentations of the Flame Princess' roll under X-in-6 with d6 skill system, or even Cadillacs & Dinosaurs 1-10 stats, make this awkward due to rounding). This works more poorly in additive systems because the opposing target number generally varies; for instance in a d20+modifier system, halving the bonus only decreases chance of success by half if the opposing target number is exactly 21.
 
In other games, opposed rolls may be handled by cross-referencing scores and the attacker rolling under the result (e.g. RuneQuest's "Resistance Table"), or by using an additive mechanic just for opposed rolls. LegendQuest and Synnibarr both use skill rating + an extra modifier (+d10 or +d20); For LQ this is done instead of the normal d100 roll (in some cases); Synnibarr makes the additive comparison only if both opponents first roll under their skill.
 
Initiative rules are particularly problematic in roll-under, since these involve a comparison between whatever number of characters (or groups) are involved in the conflict, rather than just seeing who wins out of two opponents. Attack rules are interesting; some games will handle 'defense' as an adjustment to the attacker's roll (adding more calculations to the roll, obviously, and perhaps losing some of the advantage of roll-under), while other systems give defenders a block percentage to roll against; which however means a characters' % to hit is basically the maximum possible chance to hit irrespective of how bad an opponent may be at combat.
 
Variants include
*Roll under using a number of dice varying by difficulty i.e. in The Fantasy Trip (precursor to GURPS) used a variable number of d6s (e.g. from 3d6 up to 7d6 for very hard rolls) under stat. (TFT can also handle multiple-stat rolls with more dice, e.g. unhorsing an opponent is 6d6 under [STR+DEX]. Another system for this was proposed here for D&D, where a roll against 2 stats together is made with 3d12 instead of 6d6 - this has the benefit of having a very similar distribution to 3d6).
More recently 'Summerland' used the same mechanic (with 2d6 for easy tasks, 3d6 for average, or 4d6 for difficult; a check is made against a score of 2-9 + up to two 'tags' which start with values equal to 1/2 stat).
Examples in Summerland show an issue of double counting of bonuses' where the vagueness of the rules can lead to a high or low value also giving a character a higher or lower difficulty e.g. a strong , character [high body value] getting a lowered difficulty of routine/2 dice as well as having a higher score to begin with.
A number of game systems use this for some subsystems e.g.HarnMaster combat results (# dice determined by cross-referencing attacker and defender results), as does Monsters & Slayers for some rolls (reviewed in Dragon magazine #195, Monsters & Slayers is remarkable for having different mechanics for different tasks, while still using only d6s). Original red box (BECMI) D&D suggests using this for ability checks and varies both number and type of dice i.e. either d20 or whatever the GM feels like be it 3d6 or 3d8 or 5d6 to resolve unusual situations (DM book, pg 20); Dragon Warriors also does this regularly.
This concept doesn't combine at all well with blackjack success (see next point), as higher difficulty will also increase average success result.
*Blackjack Success (e.g.  Pendragon, QAGS, 2E AD&D psionics, and 1E AD&D surprise). In games using this, higher rolls are better as long as they don't go over the skill; this lets these handle opposed rolls fairly well since the raw numbers only are compared (if A rolls 12 on the dice and succeeds, and B rolls 15 on the dice and succeeds, B wins). This is normally used with single (linear) dice rolls - in which case the blackjack method gives results mathematically the same as comparing margins of success - but that breaks down with curved rolls like 3d6.
Note that tiny modifiers on the dice can occasionally convert a best-possible ('Power Score') success into a failure, or vice versa.
Like all roll-under systems, they scale poorly beyond the point where [stat = maximum roll on the dice]. Pendragon patches this by giving characters with skills of 21+ a bonus to their d20 die roll.
Pendragon does use the same mechanic for initiative as for everything else - only one side, the highest roller, wins the combat round.
Blackjack systems are also interesting in that they tend to have 3 unusual results, rather than the usual 2. For instance in a d20 blackjack system as well as critical success ("Power Score"/rolling whatever number your skill is) and critical failure (a roll of "20"), there is a "marginal success" i.e. "it worked, but badly" result (a roll of "1"). In a normal higher-is-better system, this would be analogous to giving a special, "half-success" result on a roll which exactly equals the required number to perform a task, as well as fumble/critical, which most additive systems don't bother with.

Other Elaborations
*Runequest has ‘special successes’ and ‘critical successes’ as a proportion of total % chance, as these are relatively easy to calculate.
 
*Alternity uses a d20 roll under stat/skill, plus or minus another dice which has a size determined by task difficulty i.e. , +0, +d4, +d6,+8 up to +d20; or -d4,-d6,-d8...up to -3d20; the theory being that an almost impossible task could still succeed occasionally, or a very easy task fail, due to characters rolling a low number on the extra dice. Special successes occurred at ½ and ¼ of base success chance, making specials and criticals equally likely unless a difficulty dice came into play. (this might be less bad if there was a following 'roll' step that generated partly overlapping results e.g. if Good and Amazing rolled d6 and d8 respectively for damage, as an idea. Alternity doesn't do that).
Alternity did use the same core mechanic for initiative, with characters acting in either Amazing phase (d20 roll less than 1/4 action score), Good phase (less than 1/2 action score), Ordinary phase (less than action score) or Marginal phase (failed roll), and all actions in the same phase considered simultanous. For other opposed rolls, it assigned the defender "resistance modifiers" which applied to an attacker's difficulty i.e. a 13 Dexterity (+2 steps) might increase an opponents ranged attack difficulty from (say), a -d4 penalty to a -d8 penalty. Historical note: In the introduction to Cortex Lester Smith claims the credit for the idea in Alternity, originally proposed as a skill check variant for a possible 3E D&D, which evolved into Sovereign Stone and later Cortex
 
*TriStat dX rolls under stat with 2 dice, with the campaign determining what sort of dice these are - from d4 (normal human) to d20 (godlike). A character can buy attributes up to [2x maximum size of the campaign dice]; a 4 represents human average in an attribute, and a normal task for the campaign requires a roll of 2 dice below the attribute.
 
*Dragon Warriors is usually roll-under on d20, with defence statistics (e.g. Defense, Evasion, Magical Defense) which subtract from the attacker's stat before they roll. In the case of physical defense, this lets defense be split among multiple attackers fairly easily; the system is fairly quick since the attacker (having already subtracted once ) basically never needs to calculate how much they've rolled under by, and will knows many rolls are misses (over their to-hit score) without having to calculate.
 
*Roll Middle: beejazz proposed an alternate method for handling task difficulty, which I hadn't seen in any games in the wild. The roll must be under the trait being tested, and also over the task difficulty. The aim of this is to cut down on math. For example if a character normally needs a 13 or less to succeed (say on d20) they succeed on a 1-13. On a difficulty 3 task they fail on a 1-3 task, succeed on a 4-13, and fail on a 14-20. That trades a subtraction for a (slightly easier) second comparison, while keeping the same odds. It could be thought of as related to the blackjack method- a blackjack system often does that on opposed rolls, or if a minimum amount of success is required. It is (like a number of roll-under common elaborations) perhaps slightly unintuitive, treating numbers as individual things with separate properties rather than their being quantitative.

*a weird rule from (poster) Talysman for arrow traps for 0D&D: for an arrow trap the trap makes an attack roll, with characters who have a DEX higher than the attack roll being able to make a saving throw to dodge (folding in a 'lower is better' DEX check and a 'higher is better' attack roll using the same dice roll ?).

*In Nomine -roll-under, [generally stat+skill] with 2d6, plus a third d6 (the 'check digit') is rolled for degree of success. A 1,1,1 results in divine intervention and a 6,6,6 infernal intervention: angelic players critically succeed on triple-1 and critically fail on triple-6, with the reverse for infernal players.

*Flipping between Roll under/Roll over: from rpg.net here:  Attack rolls in Searchers of the Unknown succeed if you roll under your opponent's AC plus your own level, on a d20. Cleverly, this is inverted for sneaking around — roll 1d20 under your AC plus your level. Since AC is descending, a good AC makes it harder to get hit, but also harder to sneak around. "So in battle, wear big thick plates of metal; when stealing the king's jewels, take only your oiled-up body."

Roll-under games sometimes handle additional difficulty through extra rolls (something that can happen in any system, but is perhaps more likely in roll-under since in e.g. additive or dice-pool its easy to adjust the target number). For instance, Fighting Fantasy in one adventure (Island of the Lizard King, IIRC) lets a character 'Test for Luck' twice in order to catch a water elemental in a Pouch of Unlimited Contents)(alluded to also in monster book Out of the Pit). In this case, two rolls also double the resource cost to do this (each Test for Luck costs a point of current Luck).

These are games where dice shift types e.g. from d4 to d6 to d8 as characters get better. There's a slight problem above d12 since Zocchi dice often aren't that common; multiple dice are sometimes used instead.
D&D uses this for damage rolls; it is seen as a core mechanic in for example Earthdawn, Savage Worlds, and Cortex.
 
Simple changing-dice-type
A simple example of this would be the miniatures game Great Rail Wars (Savage World's predecessor) which is closely related to SW, but simpler; notably there's no "wild die", and no skills i.e. it just uses attribute checks to resolve tasks (stats include Strength, Smarts, Vigour, Shooting, Fighting and Guts; there's no generic 'agility'). Normal 'Gunmen' for instance might just having Shooting d6, and hence roll d6 when they're shooting something.

Savage Worlds uses a single dice for characters, plus “Wild Cards” roll an additional d6 at the same time and take the highest result, with maximum results adding and rolling again. Having only a few steps means stats are fairly granular; roll-ups make results unpredictable. It has skills rated like stats at d4 through d12 (d4-2 untrained), with raising skills above stat being double cost. Otherwise, stats don't modify skill rolls directly.

Mathematically some tasks are easier with a smaller dice (using a d4 vs. TN 5 is around 2% more favourable than using a d6), but it allows rolling for large numbers of non-wild-card opponents at once quite easily (though OTOH,  it also tends to require mass combats since single 'boss' fights will end too quickly from damage rolls blowing out). The 'wild die' both adds more of a 'curve' to results and further softens the granularity of the steps, although it does mean that for a normal check (d6, d6) the odds of having to re-roll one or more dice is pretty high - 11/36 or nearing a third of the time.
In SW, odds of a ‘fumble’ decrease as character skill increases (instead of being a fixed 5% for a d20 system lacking a ‘fumble confirmation roll’) but odds of ‘exploding’ dice (rolling up on maximum) decreases as skill goes up. Difficulty modifiers normally apply as a + or - to the final roll - Dice type is not modified for external reasons  - which can be very significant/harsh (modifiers apply to the Wild Die as well, so are more significant that step changes).
SW is particularly hit by the cap at d12; SW attempts to deliberately always be single-die so mass combats can be rolled at once, and also handling multiple dice (additive) as well as a Wild Die would be messy, so it goes to d12+1, d12+2, etc. at the highest steps.
An interesting method for fixing the exploding dice problem using FUDGE dice can be found here.
Compared to additive systems the amount of mental effort is probably less. There is an extra step of choosing the correct die, although, this is typically minimized (in combat at least) by using the same die most of the time e.g. fighting die is always the same. In SW, damage is slightly confusing in that dice are added together for this in most versions, instead of taking highest.
The Wild dice can rarely be modified e.g. Suzerain (epic SW expansion) ups these to d8s, while a Legendary edge exist that ups the wild die for a single (d12 rated) trait to d10. The CoC SW system (Realms of Cthulhu) also allows for partial wild cards, which have a d4 wild die. (Pondering as a house rule having 'untrained' checks for PCs default to just the wild die, instead of d4 with -2 to both rolls)

Other Notes
*In my own homebrewing I'd experimented with a system where stat is rated 4-12, with even numbers giving an increase in step for raw attribute checks (4 = d4, 6 = d6, 8 = d8, etc), while odd numbers instead add a +1 modifier to all the skill ratings based on the attribute (i.e. 5=-1, 7=+1, 8=+2, 9=+3) - with skills on the same 4-12 scale i.e. base 4 for an untrained skill. This involves a bit of finangling of modifiers but reduces the granularity problem a bit, while still having one-roll resolution.

*The Battleball boardgame (mentioned here) has movement rated d4 to d20, with lower being better about half the time (used to resist tackles).

*on SW's probability issue, another fix to prevent the TN problem could be to remove TNs - with opposed rolls for instance, there's no "set" TN beforehand, that would give an incentive to use d4 over d6. (That could change if a reroll is made e.g. from a 'bennie' though).

One friend's homebrew game, "Last Order" uses a single dice varying by skill (and potentially 'stepped up' by attribute bonus), with the dice size setting a 'variable raise number'; instead of each 4+ being a 'raise', raise numbers vary from 4 minimum up to about 7, with the margin being set based on an attempt to slow down the rate of several-raise results and worked out mathematically. For a normal TN 4 task e.g. a character rolling an 8 would get one 'raise' if they had skill of d4, but would have needed a roll of 11 if their skill was actually d20.
 In a sense, this gives result sort of similar to how a 'roll under' system can have higher success levels be a proportion of total success chance (e.g. like how Alternity has 'good' results under 1/2 success chance and 'excellent' results for rolling under 1/4). The system is slightly cumbersome in that of course a character has lots of different skills, and a player therefore has to record and use a different raise number for each. It also has problems should a task ever needto go back to counting success margin exactly instead of 'raises', since raises are worth more points for higher skill characters.

Combination Stat die + Skill die
Cortex uses the sum of two dice; one for attribute and one for skill. Final results follow a V-curve or truncated bell curve distribution. Unlike Savage Worlds it can incorporate d2s (since maximum rolls don't explode); it normally handles difficulty as a step up/down to the dice type.
Compared to SW, multiple rolls are slower since dice must be paired up. As a possible elaboration, one homebrew game of my design uses a similar stat die + skill die system, but also kept combat to single-die resolution like Savage Worlds by having players have to split their dice between attack and defense, taking average roll for defense. Saves and initiative are likewise kept to one die by using having them be a raw attribute check, not [attribute+skill].
A minor quirk of this setup in general [stat die+skill die] is that raw attribute checks using only 1 die- contests of strength for instance? - may be more variable than skill rolls, rather than less variable (as might be expected realistically)(the same quirk applies to dice pool games where dice pool is stat+skill, such as Storyteller). (Cortex does however often use [Attribute+Attribute rolls instead, however, e.g. Burst of Strength (Serenity pg 142) = Str+Str).
Games like this make it easy for the GM to notice and describe how much of someone's success is due to skill and how much to raw talent, if desired.

Dice Rolled determined by 'step' [dice chain]
Earthdawn uses multiple dice of various kinds additively (what is rolled being set by a ‘step number’), with individual dice able to ‘explode’ (roll up maximums). It uses multiple dice for higher steps (above d12).
Another (obscure) step die system, the 'HDL' (half dice level) system, used step numbers but with step based off [2xattribute] i.e. 2 = d4, 6 = d12, etc. and then skill as a separate (flat number) bonus; a stat bought at 1-10 set the step/dice rolled with skill score (also 1-10) then added to die roll, or raw stat rolls would roll the step and then add the 'step number' again.
 
Ad Hoc Dice Pool (dice of various sizes gathered individually).
The 2012  'Marvel Heroic Roleplaying' Game (MHR) is a changing-die dice pool where several dice of various sizes are rolled together (from different traits); 2 are chosen and added together to determine if the character succeeds, and another die used as an "effect" die to determine how well the task succeeds. A complaint on this system may be that extensively fiddly rules are needed to arrive at dice pools that feel the same in the end (with very nontransparent probabilities, to boot). Another odd feature is that the way pools are built requires several factors to be combined; something like a raw 'Strength check' isn't exactly possible because STR provides only one die. In general rolls are more likely to operate at a scene level with exactly how something has occurred requiring some interpretation. Its interesting to compare this with, say, One Roll Engine, which requires [stat+skill] pairings. Its also exactly the opposite of the original Marvel FASERIP which could apply a maximum of one attribute per check (apart from Karma effects). All MHR rolls are opposed (so both sides can roll 1s)
 
Other systems:
*Ironclaw (see discussion here) uses a take-highest dice pool, but with dice of mixed sizes. Adding a small die has little effect on chances of succeeding a high difficulty task (e.g. making a difficulty 12 task isn't helped with another d4), but increases the chances for a low-difficulty task (and decreases chance of a fumble when all dice are 1s).

*The Agora system alpha playtest (a story-game-ish freerpg by Joshua BishopRoby) was step-die, often with multiple dice but with lower rolls being better (but with the extra complication that a die rolling a '1' is discarded). (note to self- check suballocation correct)

Changing-Die/Dice Pools - # Dice and Die Type determined separately
Ork! - The RPG uses additive step dice/dice pool, with attributes rated at d4 to d12. Skills start with default rating 1 but can go up to 5 dice (using the controlling attribute die type). Dice are added directly.
Fantasy Dice (which has free version Turbo Dice available from here) is the opposite, with skill giving the base die (d4 for untrained, up to d12) and attribute giving # dice, with highest result used. Characters can reduce their pool by 1 to increase the dice one step, or vice versa. In combat characters can sacrifice dice to increase damage, or gain dice by spending their second action of the round. Characters who rolled all maximum on their dice get +1 per extra maximum rolled (something less likely as skill dice/pool increases, though as its purely numeric the shift here is probably hidden by the normal increase from pool or step gain).
(drawbacks with this sort of combined system: usually half of the game numbers are effectively on a different scale making them difficult to convert between, though perhaps its not unusual to have skill/stat on a different scale.)

Deadlands also uses a combination of changing-die and dice pool: like Ork a character's attribute determines their base die (d4,d6,d8, etc) while skill determines how many dice are rolled. It takes highest die however, with any maximum rolls (i.e. 4 on d4) rolling again and adding to that die result (exception: damage dice are added together instead of take-highest).
Unlike Ork attributes don't default to just 1d but are actually described as a full pool i.e. a stat could be 3d4 or 2d12 or 4d6 or whatever. The # dice is irrelevant to skill checks, however. Stat rolls are therefore sort-of on the same scale as skill rolls, although with the issue the extra # dice for stats is usually a wasted number - its only not-unbalanced because chargen is random anyway, as described under card-based systems).

(Another hypothetically possible variation to this might be to have have basically binary skills - perhaps just stepping up attribute dice a size - but then have multiple skills applicable to a task all be rolled simultaneously. e.g. you could imagine a 5E type system where having both a skill and a 'specialty' on that skill means rolling with 'advantage' effectively).This could work with either a count-successes or take-highest setup).

Other Systems:
*TSR's CardMaster adventure design deck, theoretically for 2E AD&D, had a stripped down mini-system of its own, where creatures rolled dice equal to level: warriors rolled e.g. d10 to hit at 7+ while monsters hit on a 6-8 on d8. This is interesting in that as well as the step-die varying, the target number is also being adjusted in parallel with it to get the results desired (thanks to Omega for mentioning elsewhere).

*Another interesting way to combine step-die/dice pool (not seen in any extant system I know of) might be to use it to represent skill 'width'. A character might buy a number of dice, with that number being say d4 if the skill is extremely general, d6 for a more specific skill, and d8 or d10 for a highly focussed specialty skill. This would balance buying skills of varying width without complex point accounting, and enables easy 'stacking' of general and specialized skills (just roll a dice pool of mixed type).

Changing Die Type - by Difficulty
Tales of the Floating Vagabond uses a roll-under step die system where higher difficulty = larger dice rolled against a stat value (pitifully easy d4, easy d6, normal d10...through to nigh-impossible d100)(it instead uses comparison of d10+stat for opposed rolls). Similarly, Greg Stolze here http://forum.rpg.net/showthread.php?603858-Games-that-use-penalty-dice/page4 comes up with a similar inverted step-die system, where larger penalty dice are bad (subtracting from the base skill+stat value).

Dungeon Crawl Classics uses an OGL-derived d20+modifiers system at base, but with Zocchi dice, and significant penalties (or extra attacks gained from level, TWF, etc) can use smaller dice such as d16 or d14, while bonuses can shift rolls to d24 or d30. Effects such as criticals are handled somewhat awkwardly due to this i.e. a combat penalty can increase chances of a critical by requiring a 16 on d16, instead of 20 on d20 (although as critical effects are generated by rolling on a separate table, which has 'no extra effect' as an option, they could have patched this by handing out a separate penalty to the crit roll). DCC is in some ways the opposite of Savage Worlds - it has dice type shifts for penalties and plusses for character skill/stat advancement, instead of different dice from the fundamental skill/stat and flat bonuses/penalties for circumstances.


General Notes: Changing die systems give a great deal of variation in ability, but regardless of rating, a high result is not guaranteed. These systems are generally fairly granular unless multiple dice are used. Unlike most single-die systems, the main way to keep opponent abilities secret is to roll behind a screen, since otherwise the dice being rolled gives away what the check is for (the same problem occurs for DMs trying to make secret rolls in AD&D, where different subsystems use different dice, but its not quite as bad). If a player gets their dice wrong, its awkward to rework the result, unlike in an additive system. Describing attributes as die codes generates some awkwardness with mathematical operations e.g. addition, halving, etc. (in part because they usually starts at d4, in part because odd-numbered dice have poor availability).
 
Varying Dice Type as a supplemental mechanism:
Shifts in dice type are sometimes used with other dice mechanics in exceptional circumstances or edge cases e.g.
*Talislanta is normally d20+modifiers on a table, but untrained rolls are d10+modifiers.
 
*SenZar uses a roll of d20 equal or over a target number of [21-attribute], but scores of 20 higher instead let a character roll a d100 vs. target 05% , giving much larger margins of success for opposed rolls (and making difficulty adjustments less of an inconvenience to these characters).
 
*Cadillacs & Dinosaurs replaces its general d10 roll under character stat with [roll of d6+damage taken] vs. stat for determining some wound effects; interesting in that they've reduced the base amount of randomness in the check to allow for extra randomness coming from elsewhere (overflowing off other rolls). This feature is probably due to that subsystem being inherited separately from Twilight 2000 (pre- the core mechanic?), rather than being a conscious design decision.
 
*The Soothsayer RPG gives stats roll-over target numbers that are usually on d10 (a stat of 3 is 10 on d10 to succeed, 4 is 9+, 5 is 8+ etc); however a score of 2 instead requires a 12 on d12, while a 1 succeeds only on 20 on d20.
 
*IIRC, 1E AD&D shifted from rolling d6 for open doors to d8 at giant-level Strength, which was a continuation of higher than the table initially went. This was replaced with a consistent d20-based subsystem in 2nd ed.

*2E AD&D's Creative Campaigning supplement has guidelines for replacing d20 roll-under ability score checks with various other dice to control amounts of variability, e.g. with Strength rolls often being low variability [d4+8], Constitution [d6+7] and Dexterity/Intelligence/Charisma d10+5 or d12+4, as well as multiple dice (Con rolls of 3d4+3).

*Basic D&D [BECMI] GM book suggests ability checks rolling under the normal 3-18 D&D stat with various dice like 3d6 or 5d4, depending on difficulty. (later skill checks for general skills in Rules Cyclopaedia are usually d20) The Immortals rules let stats go up to 100, and change from d20 to d100 roll-under for herculean, Immortal-level tasks.

*The French-language RPG Mournblade reportedly uses d10+modifiers; characters have the option of rolling d20 instead but if so, odd numbers are not added (i.e. are treated as a roll of 0).

*Fairly granular rolls such as 2d6 can be replaced with e.g. d100 tables of results to produce results changed more subtly than is possible with just modifiers to the roll (as every +1 is a significant shift here). That can be used e.g. for 2-12 attribute generation with specific races etc.

*Class Warfare for Dungeon World has a 'Herculean Appetites' special ability that replaces the usual 2d6 roll with a d6+d8 roll; if the d6 is the higher of the pair, the GM introduces an extra 'complication or danger'.

Other Notes
Hypothetically changing-die-type seems like it could be interleaved with a dice pool system fairly easily for some rolls e.g. a 'd4' could be replaced with 4 dice, count successes, or d6 with 6 dice, count successes. Perhaps this could be used where more average results are required, or where more detailed readouts of numbers are required like trying to do ORE hit locations or count 1s on a successful check, or it could be converted back the other way to simplify a dice-pool game to a single roll. Adjusting the TN here would be roughly parallel to rolling the dice multiple times, take highest or lowest.
It would also be possible to throw in a x2, e.g. 2d = d4 up to 10d =d20, given some Zocchi dice at the upper end.

With varying dice sizes: occasionally an extra bonus might be a +1 regardless of die size, or there might be a +1 that applies only to a particular die size; these can generate problems since they're bonuses that function in a 'meta game' fashion.  Complete Viking's "Lucky" gift for example (AD&D 2nd edition which isn't mostly a step-die system, but go with me here for example purposes) let a character choose a +1 on a sort of die (d4,d6,d8, etc) - this gave some reliable effects like on hit points for levelling and damage, but beyond that has benefits that varied a lot depending on which subsystems were used or on how GMs adjudicate odd situations (e.g. whether an arm wrestle is d20-under-Str or both sides rolling d4+Str score as in Creative Campaigning matters, at a wholly metagame level, to someone who gets +1 on d4s, or a +1 to d10s for hit points might vanish if a fighter was upgraded to Skills and Powers rules and goes to d12s for HPs), or (IIRC) a + on d6 or d8 might be important if using optional crit rules from Combat & Tactics.

EDIT NOTE: I have revised the terminology for this post. Where I used to call 'changing die type' systems 'step die' systems, I'm now using 'step dice' more specifically to refer to e.g. cases like Earthdawn, where a dice chain or 'step' is what determines die size.