Incorrectly regarded as goofs: Many continuity issues (hats on/off in alternating shots etc) are entirely deliberate. There are a large number of quick cuts throughout the film which serve to disorientate and confuse the viewer. The continuity in these cases is never precise and is as much a part of the final film as the script.
Anachronisms: At the beginning of the movie, Tom and Bill are going to play "checkers". The board that's used however isn't a checkers board (8x8 squares) but a draughts board (10x10). As far as I'm aware, this version of the game wasn't played in the USA in the 1930's (it was played in Europe, however).
Continuity: When Chuck takes the scarf from Grace's neck it shows up two shots later, then is gone again in next shot.