When Eve tells her life story in the dressing room, she says "then the war came, and we got married. Eddie was in the air force - and they sent him to the South Pacific." The Air Force, wasn't formed until after the war in 1947. During the war it was called the Army Air Forces.
At the beginning of Addison DeWitt's "introduction-speech" of everyone in the movie, he's supposed to look at the stage. But the rest of the audience behind him is clearly staring in a different direction.
In the Cub room, after Eve has returned from her restroom talk with Karen, Addison DeWitt is tapping his cigarette into the ashtray. The cigarette falls off from the holder. He continues tapping until he realizes it fell off and then he attempts to hide the holder behind his hands.
When Karen, Bill and Margo are returning from a long weekend in the country, they run out of gas. They mention having driven the car over the last few days, but in the two closeups of the dashboard the odometer only reads between 00023 and 00024 miles - the studio probably bought a new car and cut it apart for the interior process shots.