Donald O'Connor (Tim) wears a gold ring on his left ring finger (even though his character is not married) in almost all of his scenes. The ring is missing when he performs "A Man Chases a Girl (Until She Catches Him)", in the scene with Marilyn Monroe (Vicky) just before that, and in the film's climactic scenes.
On the marquee on opening night the Donahue daughter's name is spelled "Katie". On the poster for "34th week" it is spelled "Katy".
A poster prominently displayed announces the Hippodrome Theatre's closing "Sunday May 19th" (no year specified). The script tells us that WWII had already broken out in Europe, but the USA had not yet entered. May 19th fell on Sunday in 1940, so that must be the year they had in mind, but according to Wikipedia, the Hippodrome closed [Wednesday] 16 August 1939 and was torn down that same year.
In the "Heatwave" number, Marilyn Monroe accidentally pokes her finger in the eye of a dancer standing between the branches of a prop tree. Its pretty obvious and more so on the DVD in slow motion. The dancer jerks his head and looks down, but otherwise stays in character and continues with the number. Then Marilyn performs a twirl, sticks her head between the branches of the tree and gives him a kiss. The kiss may have been part of the choreography because she kisses another dancer before this. But it's possible it was an impromptu apology on Marilyn's part to make up for the eye poke.
Molly sings "There's No Business Like Show Business" at the show commemorating the closing of the New York Hippodrome. The theater closed in 1939. The song was introduced by [Ethel Merman] in 1946 when she debuted the role of [Annie Oakley] in "Annie Get Your Gun" on Broadway.