Other mistake: In Gene Kelly's 'Singing in the Rain' sequence, the taps he makes do not match up with the actual movements of his feet - they are much more complicated than the steps he makes. Since one can't tap dance in wet loafers, Kelly had to do the dance in the studio, and the taps were added into the picture. He apparently got a little overzealous.
Other mistake: During the car chase an Indian shoots an arrow that bounces off the car's license plate and comes back and hits the Indian. But a second before the arrow hits the Indian you can see him sitting on his horse with his eyes closed and a look of pain on his face as though the arrow had already hit him.
Revealing mistake: In the opening scene the Club's dancers are in full-swing, including the tall, lanky gentleman from Lautrec's famous Moulin Rouge poster, the one with the very angular, extended nose and chin. In several closeups it's very plain that the chin is a makeup appliance: the edge of the appliance isn't blended very well with the actor's real skin.