Visible crew/equipment: When Groucho enters the rehearsal area, the shadow of the camera is visible behind him.
Continuity mistake: During Astaire's big number "Top Hat, White Tie and Tails," he produces an invitation on stage, sings about it, then crumples it up, and tosses it away. It must be nearby on the stage, but when we finally see the entire stage, the invitation is gone.