Continuity mistake: When Jack is having his Santa joyride, the army is called in to shoot him down. There is one sweeping shot of the giant cannons rising and firing at him. If you look at the bottom of the screen, you can see two small soldiers as the guns are rising, but as they fire, the soldiers blink out of existence. (00:58:30)
Continuity mistake: When Sally is making some food for Jack while he's trying to solve Christmas the scientific way, she shoves a cork onto a bottle. The cork is shoved in far enough that you'd need a corkscrew to pull it out. When she places the bottle into the basket, the cork isn't visible at all. When Jack gets the basket, he is able to open it because the cork is out enough for someone to pull at it.
Continuity mistake: When Jack is telling Sally that he wants her to make his Sandy Claws outfit, he shows her a picture of him in a Sandy Claws outfit. We get a very clear look at the picture. However, when he is telling her which colors go where, you can see writing on the picture that was not there before.
Continuity mistake: In Christmas Town, while Jack is singing What's This, he grabs an umbrella from the snowman. Then he climbs on to one of the roofs and he leaves it in the snow. In the house he doesn't have it, but outside he has it again. (00:16:10)