Robert Patrick says in his book, "Film Moi": "The most beautiful goof in movies comes in the 'Steppin' Out with My Baby' number. Fred Astaire at one point goes into slow motion. The chorus behind him continues at normal speed, but that's not the goof. Judy Garland, in the wings watching this miracle, just beams with admiration - as well she should."
In the "Roof Garden" scene, Nadine is wearing high heels when she does the "Magazine Cover" number. In the wings at the end of it, she removes her headdress, discards her huge fan, and goes back onstage to call Don Hewes out of the crowd to dance their old number. Suddenly, she's wearing ballet slippers. In fact, in all of her scenes with Don, she wears flat shoes, so as not to tower over him.
Watch Hannah's hat in the "Fella With the Umbrella" scene. She makes it through the drops to the umbrella with her feather relatively unharmed. The shot changes to a close up and her feather is drenched. Later its standing up straight again, then down again. Its pretty obvious that the close up shots were filmed before or after the longer shots.
When Hannah is going to play "It Only Happens When I Dance With You" on the piano, she opens up the music. Music is only on the right side, but in the next shot there is music on both sides and neither Don or Hannah turned the page.
In Mike's bar, a motto is shown with the possessive form of "its" spelled as "it's". While this is incorrect today, the possessive form was spelled this way up until the 1940s, and thus could well have been around in 1912.