Continuity mistake: Hannay is talking to the milkman inside the door. Outside the door you can see across the street, the two guys walking down the sidewalk. A moment later, Hannay goes toward the door and now there's the milk wagon on the street right in front of the door. When Hannay pointed out the two guys across the street to the milkman, the milk wagon wasn't there and the milkman is inside with Hannay so he couldn't have pulled it up.
Continuity mistake: At the end of the film the Monster tells Dr. Frankenstein, to leave. The Doctor and his fiancee run off. The Monster grabs a lever and says "We stay, we belong Dead." He pulls the lever which blows up the castle. If you look to the left on the screen as the ROOF is collapsing, you can see Dr. Frankenstein knocked out on the stairs. Seconds later they cut to the SAME Doctor and his fiancee running to safety OUTSIDE the castle. It turns out in the original ending Universal killed the doc, but it tested poorly with audiences. They probably wanted a happy ending so they did a new ending where the doc lived, but they could not afford to blow up that castle again.
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.