The letter Albert writes and the one Gracie reads later are written differently, although the same words are used.
When Jerry Halliday (Fred Astaire) and Gracie (Gracie Allen) are dancing on the rotating floor, different camera angles show people in the background, but their number and position changes between edits.
This movie is based in England where vehicles drive on the left, but all the vehicles are left-hand drive, which obviously is what side they drive on in the US.
Lord Marshmorton calls Reggie a "saxaphone-playing pest", when Reggie has been playing the trumpet. However, the line is intentional, to show that Marshmorton has no interest in or knowledge of the particular instrument Reggie is playing, just that it makes noise he finds disconcerting.
When the butler brings Lort Marshmorton a glass of milk in the garden, in the last closer shot of the two, the glass of milk on the tray has been painted in frame-by-frame, probably to prevent an obvious continuity error of a less-than-full glass.
When Jerry Halliday is trying to get into the ball and Keggs is walking him around the building, Keggs says, "You wish to see Lady Alice, of course" and the shadow of the camera equipment can be seen on the wall behind Fred Astaire's shoulder.