In "Shoes With Wings On" dance, one dancer has his shoes off when trying the tap shoes. When he takes them off and gives them to Fred Astaire's character, the dancer forgets to pick up his shoes when he walks out the door. After that, they are not to be found when the routine continues.
The "Shoes With Wings On" number could not have been presented on a stage. The dancers controlling the shoes would have been visible. The "invisible" dancers could only be done with optical effects. Notice that you cannot see the inside of the shoes.
When Ezra goes to the theater to ask Dinah to appear at the hospital benefit, Jacques calls her back to rehearsal, and she says, "I'll be there in a second, JOSH" instead of Jacques.