There are two versions of Fred Astaire's "I Wanna be a Dancin' Man" number. The first was shot in front of a red curtain with Astaire in casual attire. MGM wasn't pleased with the costume, so they reshot the number in front of a different backdrop, with Astaire in a far more debonair suit. They then did a split-screen comparison of the two numbers; the side-by-side comparison demonstrated the technical precision of Astaire's dancing.
Mae West was considered for the part of Mrs. Hill but wanted too much money. Fifi D'Orsay and Beatrice Lillie were the next choices but both turned down the role, before it went to Marjorie Main.
The "walking/dancing on air" special effects were done with a traveling matte. This is especially evident when Fred Astaire is dancing alone near the top of the monument. His cigarette smoke is visible only so long as it is over his body, vanishing immediately as it moves to the side.
During the "Who Wants to Kiss the Bridegroom?" number, the "champagne" in the glasses is solid, presumably so it wouldn't spill as the actors moved around.
The ice used for the skating scenes was artificially colored to look better on Technicolor film.