Continuity mistake: During the "Why Should I Worry" song, Dodger goes with his "scarf" of sausages through a basin full of wet cement. Logically the cement should be stuck on the sausages, though that does not appear to happen, even though the sausages went through the cement. There is cement on his paws, though, that he needs to wash off.
Continuity mistake: As the movie begins, on the sidewalk in front of a store there is a box filled with kittens and a sign reading "Kitties - need home." The words "need home" are in all capital letters. In the close-up, only the "N" is capitalized and the rest of the words are in small letters.
Plot hole: When Sykes chases the gang down the subway, Fagin has to lurch to the left to end up on the subway tracks. However, Sykes couldn't do this because his car is to big to fit. Instead, his car jumps onto the subway tracks as if he were on a platform that separated the tracks, which subway tracks most certainly do not have.
Continuity mistake: As Jenny and Winston first enter the front door of their 5th Avenue flat, notice how the right-hand window significantly changes its distance in relation to the door when seen from an exterior versus interior perspective. Initially the window is one space away, but this soon changes to two spaces when Winston removes his hat and coat in the following shot. (00:32:35)
Continuity mistake: When Fagin first meets Jenny, a pail gets jammed on his right foot. However a short time later when Sykes shows up, the pail has switched feet. And at one point in the scene where Jenny reveals the piggy bank, Fagin has no pail on his foot whatsoever. (00:54:20 - 00:56:30)