In "Kidnap the Sandy Claws," just before Lock launches Barrel across the room, Lock is singing and his words are aligned. But when Barrel begins to sing, Lock's lips are moving to the words that Barrel should be singing. Barrel's lips don't move at all.
Jack emphasizes to Lock, Shock and Barrel that their mission to kidnap Santa is top secret, and no one can know. However, when they first return (mistakenly with the Easter Bunny), Behemoth is present but Jack isn't concerned about him.
After Jack orders Lock, Shock, and Barrel to kidnap Santa, they immediately set off, and by the very next scene, they are already back with a wrong catch (the Easter Bunny). Jack points them then into the right direction, and the trio start their "mission" anew. However it takes an inexplicably long time for them to return, more than 30 days, in fact. Considering that they managed to kidnap the real Santa Claus in mere seconds and how quickly they caught the Easter Bunny earlier, it shouldn't have taken them so long.
During "What's This?" Jack sings "There's children throwing snowballs instead of throwing heads". But the earlier verse ("There's white things in the air") established he never actually knew about or seen snow before.
Santa Claus prepares the presents and his sleigh first, but only then checks which children were good and naughty. It would make more sense the other way around, unless he wanted to hand out presents randomly.
When Jack's telling the people about Christmas Town, some of them keep switching places. The "Clown with the tear-away face" is in the first line at the start, then later in the second line, then back in the first line, only on the other side; the guy with snake fingers goes to the first line to the second one, then back again; the werewolf switches place with the devil, etc.
Near the end of the movie, when Jack puts a kid on the Mayor's hat, he's doing it from the left, both his arms are on the left of the hat. In the next shot, he's holding the kid with his arms from both sides of the hat.
When the cannons are being raised to shoot at Santa Jack, there are search lights in the background. When the cannons start firing, the positions of the lights jump, and they stop their sweeping motion.
When Jack falls after running into the Christmas Town sign's pole, there is a hole in the snow for his umbrella. However, he did not have it coming up the hill, as he left it hanging right after the line "the smell of cakes and pies are absolutely everywhere".
During the song "Making Christmas", some of the characters are building a ramp on which the Mayor rides the snow-mobile that Jack brought. After the small hairy Cyclops puts his piece of the ramp into its place, the 'Clown with the Tear-away Face' does the same with another piece, and the Cyclops suddenly disappears at this point. He pops up again on the other side of the screen at the end of the shot.
When Sally returns to Dr. Finkelstein's home to get her arm back, the doctor enters, as the lamp he's carrying "lights up" the entrance he comes from. However, when Sally starts back up the ramp, and Doctor Finkelstein follows, it is quite apparent that the light wasn't the lamp, itself; rather it was placed behind the wall of the ramp. This is revealed when the light casts a shadow on both Doctor Finkelstein and Sally; it would be impossible for the lamp to cast their shadows in that direction, as the light is supposed to come from that direction in the first place.
During Oogie Boogie's song, when he gets to the line "Would someone shut this fella' up?", the camera noticeably lurches a bit. You can tell, because the objects in the background shift position ever so slightly.
During the song "Kidnap the Sandy Claws", after Shock sends a cage with a bug inside of it to Oogie Boogie, you can see the wires holding up the cage as it bounces upon reaching its destination. And when Lock places Barrel onto a catapult, you can see the metal "skeleton" of Barrel cambering the puppet's feet. Finally, there is a piece of debris next to Barrel when he rises up (as Shock shows them a black box) that disappears.
As Jack ponders the holiday doors, the Easter door appears to the right of the Thanksgiving door. Later as he approaches the Christmas door, it is now to the right of the Thanksgiving door and the turkey is facing the other way (the first shot was probably flipped).
As Sally ducks behind a well to hide from Dr. Finklestein (during Jack's assigning of tasks), one of the well's capstones tilts down a bit with a smooth motion through at least a few dozen frames of this stop-action film.
When Dr. Finkelstein is making the skeleton reindeer and discovers the skull, the skull isn't really near him, as you can tell by the shadows on the back wall (the skull doesn't have a shadow). It suddenly pops into existence as the camera lowers itself (the skull's shadow is now visible).
The goof item below may give away important plot points.
Oogie Boogie has a snake for a tongue. There are some scenes when you can't see it inside his mouth when he opens it, especially in his final scene, when he is revealed to be literally a sack full of insects and falls apart - only insects are seen, no snake.