I find myself returning to the well of 13th Age again and again (and again), but as I said in my discussion of game systems I find a lot to love in the 13th Age mechanics that can be used in games that aren't 13th Age. I've also discussed in the past the variety of special dice results that exist in our games; things like the natural 20, critical failures, and the like.
13th Age incorporates a spin on the idea of special dice results with Fighter abilities that can be triggered only on dice rolls that are natural even or natural odd results, or on rolls that exceed a certain value, or when you miss. These traits, called "Maneuvers," can then be chosen during play after their die roll depending on if the natural attack roll meets the criteria for the ability. Roll an even? Call the use of "Deadly Assault" which lets you reroll 1's on your damage dice. Roll a miss? Use "Brace for It" and downgrade the next critical on you to a normal hit. How about a roll of 16 or higher? Call "Defensive Fighting" and gain a +2 AC. There are maneuvers that trigger on evens, odds, misses, values greater than X (usually 16), and even combinations like even misses, or odd hits.
These kinds of abilities make for much more interesting combat encounters, where the choice of your class powers and the value of the dice can really interact to make combat (especially for Fighter types) much more involved than just "I hit it ... again." 13th Age even has racial powers (half elf) and magic items that can change the value of the natural die roll. Did you hit, but really needed an odd value? Reduce your attack roll by 1 and gain the ability to trigger that "Any natural odd hit" ability. Fighter's never "waste" a die roll either unless they don't have an ability that triggers on that particular roll (which lends itself to making Fighter characters balance their abilities that trigger on hits, misses, evens, and odds).
In a way the Cypher System's minor and major effects are an implementation of this idea. While you can only earn a minor or major effect on a mere 10% of rolls they have a more open ended scope during play which somewhat balances out their rarity. The AGE System's Stunt Mechanic is another example of the implementation of this idea.
While this mechanic is not one that is going to lend itself to quick assimilation into other systems (being that abilities would need to be created or modified to function on these varying die rolls), it does lead one to thinking about the dice as more than just a success/failure mechanic. Gamemasters might consider implementing some similar kinds of mechanics into other roll systems. Adding a simple gamble mechanic where the player guesses even or odd before rolling and gets a bonus if they are correct for instance. Or if using something like the Escalation Die you might provide a small bonus if the attack roll is even when the Escalation die is also even (or odd when odd). These kinds of mechanics can help to take the simple and flat d20 mechanic that some find "boring" and spice it up some without adding excessive complexity. Meanwhile for those prospective game designers out there it may be something worth implementing from the ground up.