In the number "Follow in My Footsteps" Sally Lee puts her guitar on the bunk, nobody takes it but it can't be seen after the cut.
During the number "I'm Feeling Like a Million," Sally's skirt is wet after splashing in the puddles, but in the next scene her skirt is dry again.
In the final number "Your Broadway and My Broadway" there's a neon sign on the right with the writing "Broadway Melody of 1937" - the working title of this movie.
While Alice Clayton is singing "Your Broadway, My Broadway" on stage, the neon signs in the background has the actress's name ('Sophie Tucker') on them.
When Eleanor Powell overhears Robert Taylor and Robert Benchley discuss the show's investors pulling out, the left "tail" of the bow on her dress (viewer's right) is bent over. When Taylor goes to her for the closeup, it is straightened out. When they both leave through the stage door and stop to talk outside, the bow is folded over again.
There is an audible change in the quality of the soundtrack in Judy Garland's "You Made Me Love You" number when she goes from the spoken interlude to the reprise of the melody. It appears that she pre-recorded the sung portions but delivered the spoken parts "live" as the scene was shot.
After meeting everyone in Alice's boarding house, Steve and Sally go up to the roof and look down at the street, which Sally identifies as Broadway. Since the boarding house's address would have it near the intersection of 9th Avenue and 45th Street, it would be impossible to see Broadway from the roof.