Audio problem: When Gene Kelly starts to walk away from Debbie Reynold's car and his coat rips, he becomes stuck for a moment about halfway through so the movement is rip - pause - rip. On the soundtrack, however, we hear one continuous ripping sound.
Continuity mistake: After singing "Good Morning" Don, Debbie and Cosmo are sitting on the couch laughing. Debbie leans her head on Cosmo's shoulder, then we cut to a closer shot and she's suddenly holding her head straight up.
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.
Visible crew/equipment: In the "Good Morning" dance sequence when the trio exit the kitchen and move into the living room, the plywood wall separating the two rooms is visible so it is clear that this was shot on a stage rather than in a house.
Visible crew/equipment: During the Broadway Melody, the shadow of the hanging boom mic becomes visible. This happens during the sequence filmed in the large sound stage. Kelly is dancing with a woman with a long fabric flowing behind her. The shadow is on the right side.
Suggested correction: I can see no shadow at any point in this scene. Could you provide a time stamp or screen capture? Besides, there is no spoken dialog or even any natural sound in the scene. Why would they be using a boom mike?
It's on the right, center side of the screen when the lady with the long flowing dress is further away and the shot cranes up. This is a musical number, Kelly is singing. They would still use a microphone.
Are you talking about the fantasy sequence between Kelley and Cyd Charisse? If you mean the faint shadow at 1:25:18, that is a shadow from the flowing cloth. And no, they would not be using a microphone. 1) There is no singing in that sequence. 2) In musicals at the time, all singing was pre-recorded and the actors lip-synced to a playback on stage.
Other mistake: Don, Lena, and crew have a nearly impossible time when starting to shoot "The Dueling Cavalier" as a talkie. But, before that, we see a different group shooting the complex, Zigfeld-esque "Beautiful Girl" number for another picture with no problems at all.