Revealing mistake: During the opening musical number, right before Gary joins the large group of people for the dance, he accidentally kicks Walter out of frame. Watch closely though. His foot comes nowhere near Walter when he supposedly "kicks" him.
Visible crew/equipment: At the very end of the movie when they are singing in New York, you can see the camera crane for about five seconds right before the credits.
Audio problem: In Notes II, Madame Giry voice is out of sync as she says,"Fear his fury.".