When the Polish squadron of the RAF finishes singing; Jose Ferrer takes off his glasses. His glasses have flexible temples (ear pieces), that wrap around the ear. About one minute later he puts on glasses that no longer have the flexible temples; but instead have fixed type (the type that is common since the war ended) that just have a slight bend in them.
At about an hour in, a flipped shot of Schultz (Christopher Lloyd) puts his armband on the wrong arm.
After the air raid two of the actors are cleaning up outside the theatre. Mel Brooks arrives in a car pulled by a horse with reins clearly visible. In the next shot, close to the driver's door, the reins are no longer there.
When Anna looks in the mirror while Siletski is on the phone; her left hand is underneath her purse holding it. However, in the next cut, her left hand is now on her stomach while she uses the rest of her arm to hold it. After that, the orientation of how she is holding the purse keeps changing with each cut.
When Frederick is in the basement with the rest of the company taking shelter and a bomb hits; his left arm is on Anna's chair and his right is in front of him. However, in the next immediate cut, his right arm is now resting on the globe.
In the introduction it is stated that all of Czechoslovakia was absorbed by Nazi Germany. In fact Slovakia was constituted as an independent regime that fought in WWII as one of the Axis allies.
Kinski displays a pink triangle on his lapels when going out, explaining this to be mandatory for gay men. Even though Jews were forced to wear the infamous yellow badge in public in occupied Poland after November 1939, gays were not subject to this practice of public discrimination. Pink badges were used exclusively inside concentration camps.
Hitlers personal plane was a Ju-52 not a Ford Trimotor.
The Piccadilly Theatre is spelled "Picadilly".
Anne Bancroft's character's name is Anna Bronski. However, it should be Anna Bronska because it is an adjectival surname and all adjectives in Polish have masculine and feminine forms. The endings -i and -y are masculine, the feminine equivalents of adjectival surnames end in -a.
Obvious stunt doubles for Mel Brooks and Anne Bancroft during the opening musical number when she twirls and both of them are flipping one another over each other's backs.
Bronski becomes agitated when Sobinski wishes him good luck, because wishing an actor "good luck" is bad luck. Yet earlier, Anna wishes him good luck and Bronski is not at all bothered.
Although having Anna give the cue line "To be or not to be" to the men in the audience she wishes to meet in her dressing room, is a very funny premise of the film, it actually would be highly impractical for Anna to think she would have time to meet backstage. Hamlet's "To be or not to be" soliloquy is only about 3-4 minutes long and Ophelia has the very next line in the play, (in fact Hamlet announces her entrance at the end of his soliloquy) which would barely give Anna any time to meet men in her dressing room during this monologue.
In the credits, apart from "Picadilly" being incorrect there's a typo near the end: "Title Illstrations".
In the Naughty Nazis song, "A Little Piece...", they mention Pakistan as one of the countries. Pakistan was not created until 1947.
When Mel Brooks character's car is being pulled by the horse you can see a line of some sort that is being used either to pull the horse forward or to help the horse pull the car.
On his way back to Poland to find and kill Professor Siletski, the plane flies through a region with high mountain areas, and when Lt. Sobinski jumps, his parachute unfolds, while in the background the same scene is shown. There are high mountain ranges in Poland, but they are far from Warsaw, where the Lt. said he would go as quickly as possible in order to prevent the Professor from reaching the Gestapo with his list.