Continuity mistake: When the hermit teaches the monster that fire isn't always bad, he has a cigar in his mouth. When he gets some firewood, the cigar is suddenly in his hand.
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 of the screen as the roof is collapsing, you can see Dr. Frankenstein knocked out on the stairs. Seconds later they cut to the 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: When The Monster is roaming in the woods early in the film, he comes across a shepherdess and her flock of lambs. Having fallen in the stream and been pulled out by The Monster, she starts screaming. In the long shots, The Monster puts his hands across her mouth very quickly to silence her, but in the close up shots of her, his hands are missing, even though they keep switching between the two shots.
Other mistake: It's the scene when Dr. Frankenstein's fiancee is being held hostage. To prove that she's all right, her captor allows her to talk to Dr. Frankenstein through a telephone type device. She's supposed to be tied up with her hands behind her back. But she pulls one arm from behind her for a fraction of a second, then tucks it back behind her again.