Other than false swipe, hold back, splash or celebration
This would really only be possible against a Shedinja with a Focus Sash, if you used a supereffective attack against it, for example Flamethrower. It would survive without taking damage whatsoever. That's the only situation I can think of.
Hope I Helped!
If an attack would deal an exact multiple of 65,536 damage (but not an exact multiple of 1,048,576), the damage ends up overflowing to 0, and does so at a late enough point that the game doesn't find it when it checks for "This damage is less than 1--I'd better tick it up to 1!"
I engineered the setup for https://www.youtube.com/watch?v=FhjwUJksXv8 and gave it to the two battlers who showed it off there, which was carefully calibrated so that all the possible random variations in damage end up producing exact multiples of 65,536. It's possible to do a similar setup in earlier games, such as the ones that existed back when this question was asked, but the numbers get more complicated (such as this Showdown replay).
If you use a move like Fissure or Sheer cold on a Pokemon with Sturdy, it wont do any damage, but will still waste a turn and PP.
Mimikyu with Disguise on the first turn that Mimikyu is present it will protect the first attack of the opponent and then it will be vulnerable.