Super Mario Bros. 3/Glitches

Note that the glitches here apply to the NES version, and many of these have been fixed for later re-releases such as in Super Mario All-Stars or Super Mario Advance 4.

Gray Hammer Mario
Enter the bonus room in 7-8 which includes a Hammer Suit as Tanooki Mario. Jump at the giant ? block, then stand so that the bouncing suit hits you, but transform to a statue before it hits you. If you did it correctly, you should transform into gray Hammer Mario. You're completely invulnerable in this form - but you cannot enter the pipe to exit this room, so you're stuck until time runs out.

Pipe bug
Go to either 3-9 or 7-5 as big Mario. Both have pipes directly beside ice blocks. Stand on the pipe and remove the bottom-right ice block from Mario's position, then run and duck to get inside (kick the ice blocks under you away as well). Once you're in this small gap, duck and jump to try to wedge yourself between the ice block and the pipe and hold down. If you did it correctly, Mario should eventually go down beside the pipe and land on the other side, except he's nowhere to be seen. You can now move the screen freely, but are stuck here indefinitely. You'll have to reset to get out of this.

Unfiery Bowser
Enter Bowser's Castle with a P-Wing. In the large room with the donut blocks, select the bottom path, and play all the way through until you land in Bowser's chamber. Here, fly all the way up and to the left to see a brick wall, fly over it then return back. Now drop back down to fight Bowser and he won't shoot any flames for the entire duration of the fight.