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.
Audio problem: When Donald O'Connor walks onto the keys of his piano at the beginning of his Make 'Em Laugh Sequence, the sound that we hear is a chord, but in reality we would hear a much more dissonant combination of notes, because several notes adjacent to each other would be played at the same time.
Continuity mistake: After Kathy and Don find each other again, and he's trying to tell her how he feels for her, they are walking outside on a sunny day. He opens a door to go into a set, but when the camera angle changes, it appears as though they are walking from a dimly lit hallway. (00:41:30)
Continuity mistake: During "Make 'em Laugh" Donald gets flipped off the board, and sings on the ground. There is a man sweeping next to him. You see the broom coming in and out of frame near his feet. When the shot goes long, the sweeper is behind Donald, and no one is next to him. (00:28:15)
Audio problem: When they are filming the "Duelling Cavalier" with sound for the first time, when the operator says that "She's gotta talk into the mike. I can't pick it up." his lips aren't moving at all. (00:49:33)
Revealing mistake: At the first premiere (at Grauman's Chinese Theatre), look closely at the audience. Because at the end of the movie, and several months later at the premiere of The Dancing Caviler, it is the same audience! (Same clothes and even the same seats!)
Continuity mistake: When Don Lockwood is telling R.F. about the Broadway scene, when he says "It's called Broadway Melody," he has his hands right in front of him and he's looking at R.F. In the next shot he's looking down at his hands and then turns to R.F. again and puts his hands up in front of him. (01:15:20)