It becomes fairly obvious during the "Fated to be Mated" duet between Fred Astaire and Cyd Charisse that Charisse is wearing a skirt one moment and culottes (or flared shorts) the next. The bottom half of her costume changes on each cut of the dance when they are doing deep knee bends, and this is where the culottes show. For the upright spins and lifts, the skirt shows. The dance was obviously performed twice and edited into one sequence.
When the Former Commissar of Art gets sacked, there's a yellow lamp hanging over his desk. (He actually walks behind it.) Then, when Vassili Markovitch answers his phone, the lamp has disappeared and been replaced by Vera.
When Peggy Dayton first gets off the plane in Paris, she is handed the same bouquet of red roses twice.
During Peggy Dayton's song Satin and Silk, a long shot shows her moving away from Peter Ilyitch Boroff, who has his hands in his lap. After a cut to medium, he suddenly holds them up to his tie.
At the top of the screen, when Brankov, Bibinski and friends visit Ninotchka in her "section" in Russia.
When she's typing on the typewriter in the hotel room, Nina presses only the same two keys.
A man using a flame cutting torch is wearing an arc welding helmet.