The position of hat on the Congeroo scene drummer changes between close up and group shots. Around minute 50.
Martha Raye's shoes change during the play-within-a-play scene where she is suddenly singing on a treadmill. For some shots, she's wearing silver strapped heels, but in others she's wearing women's tennis shoes, and still others she's wearing some sort of slippers. (Women's athletic shoes in the 1940s seem odd to modern audiences, as they still look like dress shoes by today's standards. Despite the rubbery sole and high ankle protection, women's athletic shoes still had heels and were shaped like low-heel dress shoes.)
Betty picks up a rifle with a bayonet attached, but in the next shot it's a double-barreled shotgun.