In the last Christmas Carol scene with the group of carolers, the girl on the far right side is seen in different arm poses. Thru 3-4 cuts, you can see her hand near her neck as if she's holding onto the collar of her shirt, the next cut, she is crossing her arms. This happens about 3-4 times. We can see there were obvious cuts in the filming.
When the squeegee man is saying "honest living" repeatedly during the musical number "Christmas Bells," he underestimates the number of repetitions and has to catch up to the voice track.
When Collins is holding the long note towards the end of the musical "I'll Cover You (Reprise)," he stops short while the voice track continues on.