The strangest pause bug I know is in Mario Sunshine: pausing will misalign the collision logic (which runs 4 times per frame) and the main game loop. So certain specific physics interactions will behave differently depending on how many times the game has been paused modulo 4.
really? is one state the one where you fall through bridges? I can't play Sunshine because of that.
There is another great one in one of the Legend of Zelda games.
The game world is paused whenever Link pulls an item from a chest, but because his animation does not loop perfectly, because of a missing frame, he slowly slides across the ground and even through walls.
One of the minimum % speedrun abuses this by looping the animation for many hours in order to glitch through a wall, and not collect a progression item, which would count towards the collection percentages.