Continuity mistake: Wiesler is called into the office and his boss opens the humidor to get a cigar. As Wiesler is talking you can hear Grubitz lighting a match. The next shot his left hand is nowhere near his mouth and his right is holding the cigar in his mouth and is already a bit smoked. This would be impossible as if he used his left hand it would have been seen coming down after he lit the cigar. Even if it was his right hand then how did he get it out of his hand so quickly and then hold the cigar all at the same time.
Continuity mistake: While Dreyman is meeting Jerske at his apartment, Dreyman is smoking a cigarette. As Jerske says, "Happy author can write," the shot shows Dreyman at the same time and smoke from his cigarette coming from holding it in his right hand below at least his chest. The next shot, Dreyman is holding the cigarette in his left hand and smoking it.