During the "Where Is The Life That Late I Led" number Howard Keel leaves the stage and walks out on a long runway that extends into the audience. This runway is not visible before or after the number though there are several shots from the stage into the audience.
When Bianca is by the fountain, she is carrying a net which she tosses away. It lands to the right of the fountain. A few shots later the net is missing; a few shots after that, it has returned.
In the performance of KISS ME, KATE, the women wear high-heeled shoes as part of their costumes. While women of the Middle Ages wore slippers as opposed to shoes with heels, this is simply an error of the producers of the play within the movie; it may have been intentional by the movie's script writers.
Boom mic shadows are visible in scenes backstage during the performance of the show.
At 00:44:30 during the "Tom Dick and Harry" number, Bobby Van (in the purple suit) trips on Bob Fosse (in the red suit) and breaks character and looks towards the camera and crew as if waiting for the director to yell "cut". This occurs to the right of the screen.