Revealing mistakes: When the sliding upside-down car finally comes to a stop on the frozen lake, you can clearly see that there's no one inside it, which conflicts with the close up interior shots of Monsieur Septime and the secretary.
Audio/visual unsynchronized: Throughout the movie, it's obvious that the pianist in the restaurant isn't really playing the music that we hear.
Continuity: During the dance rehearsal, the broken plate on the floor changes between shots.
Continuity: When Septime is being instructed to walk around the news stand, a man and woman approach the stand twice between the changing shots.
Continuity: In the chase scene when Septime drives his car into the snowbank, its on an angle, but the next shot shows a car shaped hole that is shorter than it should be and as if the car hit straight on.
Continuity: As Septime and the secretary tumble down the hill their clothes get covered with snow, but when they reach the bottom they're clean again.