Final Fantasy Legend 2 (SaGa 2 in Japan): One that I stumbled across by accident (but is well known) is the trashcan bug. Basically, at certain MAGI counts (those that correspond to the indecies of items usable outside of combat), you can use the trashcan as an item outside of battle and it will do something. However, it turns out that using that uses up your Power MAGI. Eventually, when you run out of Power MAGI, your MAGI count will become 99 (actually 255), allowing you to skip part of the game. Combined with another bug I read about years later, the Dragon Warp (which allows you to take a dragon outside the race, at which point you can fly it through walls and other obstacles), this allows you to skip from the fourth world straight to the end of the game (use the Pegasus MAGI to escape the Dragon Race on your dragon).
Of course, there are some catches:
* If your MAGI count becomes 99 (actually 255, but the game only displays 99), collecting another MAGI will overflow the MAGI count to 0. This is a softlock unless you do this in Monstrous Dungeon (where you can use the Dragon Warp glitch to get out of it) or later (when you're past the doors that check your MAGI count).
* If your MAGI count is 255, your inventory is full, and an enemy drops an item, the item's index will overwrite your MAGI count. This is fine if the item is Punch (which has index 79, which is more than the number of MAGI available in the game), but not so fine if the item has a lower index (I was actually softlocked once because of this).
No screenshots here, but I think a description is better than a screenshot in this case.