Personally, of the dice mechanics I know of, FATE is probably my favorite, followed by d20.
Fudge dice(FATE)
Description Effectively 4d3-8, you roll 4 d6s, each of which has three outcomes, +1, 0 and -1. All the dice combined + your roll modifier sets it.
Advantages Strong bell curve that stays between -4 to +4.
Easy to buy. You can just use d6s if you can't find fudge.
Disadvantages Slower than 1 roll systems, you need to add it all up.
Potential swing is larger than most modifiers in the system. This is mitigated by the strong bell curve of the system of course.
d20 rocks on 1 roll, which makes resolution very fast, but its completely linear and thus swingy. Skill contribution varies by level. Mostly valued for speed of resolution and easily estimated probabilities.
WoD d10. Hideously swingy AND slow. You roll a number of d10s equal to your dicepool(two numbers ranging from 1-5, subtract circumstantials from the pool before rolling), d10s that get above a certain value(8, 9 and 10) count as successes. 5+ successes is effectively the critical. So its swingy, biased towards failure, since you can roll 0 successes even with 10 dice(most talent + most skill). Slow too, because of the number of comparisons you have to make per dice(form dice pool, ask for penalties and subtract, roll and roll explosions). And it explodes. Mitigation allows rerolling, but not gaining success.
Exalted d10 is somewhat similar, except you can buy successes, and it explodes by simply doubling successes on a 10. Additionally, the subtraction step is removed, you roll against static target number of successes. This speeds it up, but makes it harder to meet targets consistently without purchasing successes.
System independents:
Opposed rolls -Never liked these in any system, they massively overrate luck's influence on the exchange.
Exploding dice - Much like critical failure tables, these make fun stories once in a while, but they generally do little more than waste time of resolution, and add more swing.