On the occassion of the silver anniversary of Super Mario Bros., Nintendo has put out this video celebrating the most famous glitches of the original, ones that first spread not by Internet or message boards, but grade school lunch tables.
Some of these more qualify as exploits (the infinite 1-Up trick) or novelties (dancing on the beanstalk), but that 3-4/4-1 tiny fire Mario is both of those and a glitch. It's certainly useful - watch as Mario slips beneath the Lakitu before leaping up to fireball it.
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The subterranean Koopa shell, sliding underneath the pipes, is another classic exploitable.