Our group has been taking issue with traditional systems for rolling and resolving skill checks/skill encounters, and we were wondering if there are more dynamic and realistic options out there.

Most games (in my experience) are some variation on [skill + misc bonuses + your roll vs. difficulty] trying to roll "high" (or low, same thing really). Even GURPS falls into this category ([roll vs. skill + difficulty]). I understand that this model is simple and adding bonuses and penalties and getting clear outcomes is really easy, but there are major problems with this model:

It's Swingy. Sometimes, regardless of your skill level, you roll great, or terribly, and this may happen consistently or randomly, and you get situations where low-skill characters consistently roll high, high skill characters consistently roll low, etc., and so much of the time your skill checks (especially with plot-relevant skills like diplomacy) are really crucial.
Bonuses and Penalties Don't Actually Matter as Much as Rolling High. This is an extension of 1). Take DnD, or any other example like it: What does a +1 really mean? All it means is that, in the exceedingly rare case that you missed your roll by 1, instead you pass. +2 means that if you missed your roll by 2, you pass. Etc. In order to have consistent effects from a skill bonus, that bonus has to be pretty high, to the point where you just start auto-passing stuff, and then the DC gets raised, and things are still just as swingy.

Solutions that Don't Work

Some people have tried to get around this by building in a re-rolling mechanic, and others (like Steve Jackson) have brought in a bell curve for their die rolls. I think these solutions are real attempts to solve the problem, but they seem to me more like curing the symptoms and not the disease, which sort of leads me to the question, "Why are we rolling dice at all?" or "why are we using dice in this way for skills?". I know things like Amber's Diceless attempts to do away with the whole problem, and I'm not really convinced that that's the right option (I'm open to hearing arguments, though), since it precludes all the excitement and fun of Skill Challenges (like DnD 4th, which we really liked).

What We Do Want

The ideal skill system,

Doesn't have the above problems (swingy, bonuses and penalties don't matter as much as rolling high, isn't just a "patch" over this kind of broken system).
Can handle simple, non-critical skill checks without much fuss or fumbles or absurdity (i.e., cooking a meal, walking a tightrope, piloting a ship).
Can enter into robust skill challenges (a car chase, heated negotiations, a formal debate, trying to out-hack another user on a network) that are a substantial as combat and aren't just a series of swingy rolls.
Is pretty clean and elegant, without a ton of crunch.

Any recommendations?

Clarifications and Definitions

A couple folks have asked about some clarifications on these terms.

"Realistic" doesn't translate to "exactly how it would be in real life, down to the precise percentage." Because roleplaying games are simulations and only aim to present a model of reality, a system where character aptitude varies wildly and arbitrarily regardless of skill doesn't model the world we live in very well.
"Robust"/"engaging"/"dynamic" just means that the system can actually handle describing a level of intricacy and complexity in character actions. Compare "Roll a diplomacy check to see if you convince him" to DnD 4th edition skill challenge with variable actions you can take such that there are different strategies and variable outcomes other than binary pass/fail.
"Not a ton of crunch", as someone pointed out, is somewhat contradictory with 2. I understand that this is subjective to your level of comfort/love of math, but I think a good definition of "too much" crunch is where the calculations and steps are too numerous and/or too complicated either for the characters to easily grasp strategies about how to proceed, or playing the game is bogged down by what is essentially a convoluted mini-game that distracts from progressing the action of the game.

Hence, the ideal skill system falls somewhere between the extremes of 2 and 3 (one-shot resolution with no strategy other than rolling high and a highly convoluted system that bogs down the entire game).

PS. Don't worry about recommending a whole system, we're just looking for recommendations for a skill system. Homebrew solutions welcome.

Best Answer

Try Fate Core.

Fate (Core and Accelerated) is available for free download here and are online here. It's a setting-agnostic engine (invent your own setting in collaboration with the group), which uses a small set of skill-based options to resolve nearly every action.

Swingy: No. Fate uses Fudge dice which provide a strong bell curve between -4 and +4, with most raw die rolls being between +2 and -2. There is also a currency of narrative control (gain it when bad things happen to your character, spend it to help your character succeed later) which can be used to modify rolls.

Bonuses and penalties don't matter: No. Because the majority of rolls are only 1 or 2 in either direction, this makes skill modifiers the most important part of determining success in a character's specialisations. A character's highest skill is usually +4 at the beginning of play, which turns even the lowest possible die roll into a mediocre success.

Non-critical actions are handled simply: Is built into the system's ethos! You only ever roll the dice if both success and failure would be interesting. Otherwise, just go with whichever is interesting (often failing a simple check results in just trying again, or in the story coming to a halt--so just assume success and move on).

Robust skill challenges: Are part of the system. When you're trying to achieve something, it can range from a simple opposed roll to a contest (everyone makes a series of non-opposed rolls against the same target numbers and the person with the most successes wins), a challenge (everyone makes a series of opposed rolls to compete for success), or a conflict (opposed rolls in a more "traditional" combat system style). Each is well-suited for particular kinds of opposition and the choice is always driven by the narrative. Regardless of the mechanical framing used, resolution of opposition is often much less clear-cut than "win and lose," as "success at cost" is a common option for avoiding total failure, while the system's narrative currency means failure is designed to contain the seeds of future success.

Clean and elegant, not a lot of crunch: Yup! Aside from some default examples, Fate Core is an engine that encourages players to design the level of crunch they're comfortable with for their own characters, and gives clear guidelines on how to do so without major imbalance. Fate Accelerated is a 75-page pared-down version of the engine, while Fate Core includes more "dials" to turn to the complexity your players want.