In Chapter 10, when the Spy (Gene Garrick) is caught listening (using a hidden bug microphone) in on the Commissioner's meeting, he is taken into custody. The Commissioner then tells the arresting officers "a job well done" into the bug mic. There is no way he would know they arrested the Spy.
Roughly 12 minutes into chapter 4, there's an enormous hair in the gate in the lower right-hand corner. As it comes and goes, it was obviously there when the scene was filmed.
The Nazi leader seems to take forever doodling a simple skull and crossbones sketch. He's drawing it through the entire movie.
Every time the leader points to a place on his globe he uses his middle finger. This should have been caught by censors as a obscene gesture.
In Chapter 6, the boom mic shadow can be seen moving around behind Dr. Elise Bork as she enters the room.