The part where Sally is worriedly talking to herself while Jack is practicing his "ho ho ho" in the background.
"A pox? How delightful, a pox!"
"If you please..."
The street band attempting to play "Jingle Bells", with tritones for the harmony, and later their rendition of "Here Comes Santa Claus" in the same style. Evidently, those guys do not know any major keys.
When Lock, Shock and Barrel try to stuff Santa Claus down the chute leading to Oogie's lair, he gets stuck.
Shock: "I think he might be too big!" Lock: "No he's not! If he can go down a chimney, he can fit. Down. HERE!"
And when we see an outside view of the chute as Santa grunts and cries out as he goes down it, knocking rivets out on the way.
And also right after they first shove him in the opening of the pipe and realize he is stuck we get a shot of Santa's mistletoe underwear. How humiliating for Santa.
As Santa is shoved into the opening of the pipe he makes a funny face by rolling his eyes in two different directions.
The almost shit-eating look of pride on Jack's face while the Mayor gives a speech before his Christmas voyage.
Near the beginning of the movie, Sally detaches her arm in order to get away from Dr. Finklestein. When he yells after her, the arm left in his hand starts bonking him on the head.
Dr. Finkelstein kissing his new creation... and the top of its head promptly slamming down on his face.
In a deleted scene, we have Finklestein's escape after he's unmasked as Oogie Boogie.
Finklestein: (being carried into a chamber) Farewell, Jack (falls) SKELLINGTON! (door slams on him)
A brief one, but Jack's expression when Santa Claus flies out of Oogie's lair when all is said and done. You can practically hear him saying "I didn't know he could do that."
One of the witches uses her hat as a witch trumpet.
At the beginning of "Jack's Obsession", the werewolf can be seen doing a small pivot while singing.
Jack's unexpected Deadpan Snarker moment while borrowing equipment from Dr. Finklestein:
Dr. Finklestein: Curiosity killed the cat, you know.