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!)
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: In the scene where Don Lockwood is dueling in "The Royal Rascal" at the beginning of the movie, when Don opens the balcony door and the other dueller comes after him, Don flips him over the balcony and he splashes into a puddle. Meanwhile, two guys come out after Don. The next scene, one guy is lying on the balcony, Don jumps over the balcony landing on his back, and there is no puddle, no guy lying on the ground, but a bushy garden instead.
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.
Continuity mistake: There are two continuity mistakes when Don Lockwood makes his escape from his fans, and jumps onto the passing trolley. He climbs to the trolley roof and walks along the roof to the back of the trolley, when he jumps off to land in Kathy's car. However, in the next shot he's jumped off the front of the trolley, not the back. Also, in the distant shot, Kathy jumps as he lands in her car, but when the shot changes to a close-up, she's calm and then jumps a second later.
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.
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.