The position of the sleeping children in the bed changes between shots a couple times.
At the police station, Inspector Bailey removes his glasses, folds them, and begins to put them in his inside coat pocket. In the next shot he is unfolding them and puts the glasses back on.
The lights above the elevator on the ground floor of the United Nations building indicate that the elevator travels 36 floors in a few (i.e., 3-5) seconds. That kind of acceleration, speed, and braking would injure occupants of the elevator, especially the elderly operator. That distance in that period of time would exceed 60 mph.
During his first solo, trombonist Jack Teagarden's hand movements don't match the music on the soundtrack.