A Hard Day's Night (1964):
It's a bit tough to see, but it's there - the Beatles wear different clothes in different shots as they're chased by fans in the film's opening sequence. This was owned up to by none other than the film's director, Richard Lester, in You Can't Do That! the making of A Hard Day's Night. Since the Beatles had never made a movie before, they didn't understand about continuity. For part of the scene, they were in their costumes; for others, they were in the clothes they came in with. The continuity supervisor had been in the bathroom when all this was happening, and she sent Lester a note announcing her intention to quit if the rest of the movie was going to be done in the same way. (It wasn't, and she did stay on until the end.)
History of the World, part I (1981):
During the "Spanish Inquisition" scene, a group of synchronized swimming nuns come out of the water with sparklers atop their heads. This effect was achieved by lowering them into the water, then reversing the film. Watch the smoke, and you'll see it's billowing towards the sparklers, instead of away.
The Hunchback of Notre Dame (1996):
In the Disneyfied version of the classic tale, when Esmerelda (v: Demi Moore) cuts away the ropes from around Quasimodo (v: Tom Hulce), all of them get cut. However, in the next shot, there's still a rope around Quasi's neck, which Esmerelda tosses away a few seconds later.
Also, Phoebus (v: Kevin Kline) is shot in the back, but his wound is on his front. (And it isn't an exit wound, because his back is uninjured.) Plus, the tear in his shirt fixes itself.
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