Continuity mistake: In the scene where Harpo is climbing out of the brig and goes to the open window of the famous aviators, he is shown climbing in the room from the outside with the rope he's climbing on draped across his chest. When he gets into the room the rope has disappeared!
Continuity mistake: During Astaire's big number "Top Hat, White Tie and Tails," he produces an invitation on stage, sings about it, then crumples it up, and tosses it away. It must be nearby on the stage, but when we finally see the entire stage, the invitation is gone.