Not really an exploit, but I did exploit it alot! In Professor Layton when I got stuck on a puzzle, I would save my game, use all three hint coins and then load my game. I know, not a glitch, just plain old fashioned cheating! There was a glitch that I could do in the aquatic ruin zone in Sonic 2. There was a certain ledge that if you jumped at just right, would get Sonic stuck inside it, upside down. You could turn left, right, duck and look up but only get out when you jumped. I have never been able to duplicate this on an emulator and I no longer have a genesis. Not really useful, except that none of the enemies in the level could reach you there.
Any and all glitches from Super Metroid, especially the speedball ( rolling in a ball at the same speed that samus runs)
Its best on the easiest level, on the level inferno, I believe. I tried single handedly getting the Genocide gun, but I think its impossible unless you grind like a madman...and are still pretty good at the game to boot!