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.
Other mistake: There's a minor technical error in the portrayal of the telegraph equipment at the train station. When a message was coming in for the stationmaster the repeater relay, on the left, was shown clicking away. The telegraph sounder to the right wasn't moving. Both should have been in use as the repeater relay would have been fairly quiet while the sounder would have created the loud audible clicks. Repeater relays with fresh sets of batteries permitted the telegraph signal to travel greater distances by regenerating the Morse code signal at each telegraph office. The sounder made it much more audible. An office at the very end of a telegraph line would only have a sounder.