Crew or equipment visible: Spectators can be seen watching the filming of the "New York, New York" number in Rockfeller Center (though it could be argued that the sight of three men in navy uniforms singing and dancing might attract attention, even in New York).
Crew or equipment visible: Near the end of "Prehistoric Man" when the dancing moves in front of the dinosaur, Claire's mark can be seen on the floor.
Plot holes: When the boys are looking for clues on the poster in order to find Miss Turnstiles they find her likes and dislikes. The only problem is none of that is actually mentioned on the poster they have or any that the viewer sees.
Continuity: Early in the movie, Gene Kelly accidentally breaks the glass partition in the taxi between the sailors and the driver. He puts a crack in by hitting the glass in excitement. A few scenes later, the glass panel is restored.