In the beginning of the movie, when Danny is getting gifts from his friends in Kokomo, he is already holding the sheep's skin before he even gets it.
When stage 13 gets flooded, it's Friday at 3pm. Then it goes by one day and they say the premiere of Lil Ark Angel is on Friday. But it must be a Saturday.
When Sawyer is explaining to Danny about how people don't care about the animals talents in Stage 13 after the rehearsal, during the conversation the shadow on Sawyer changes from curved in one frame, when it cuts to Danny and back to her, it is then straight.
The movie-within-the-movie, Li'l Ark Angel, is a Biblical epic about Noah's Ark. Such a film would not have been allowed to be made in 1939 as it would have violated the Hayes Code, a notoriously strict set of rules that all major Hollywood studios had to follow at the time.
In the beginning of the movie when Danny receives his "script", it clearly says at the top that he is "Cat #1". However, when it comes time for the animals to sing the noises they make, Sawyer sings her sound before him, making her "Cat #1".
As Danny arrives in Hollywood at the beginning of the film, the Hollywood sign can be seen. The film is set in 1939 so the sign should still read Hollywoodland as it did until 1949.
The Lil' Ark Angel premiere is being projected on a 1.85:1 screen, even though the film was shot and released in the late 1930s, and all films at that point were shot in Academy Ratio (1.37:1).
As day breaks at Grauman's Chinese Theatre, the Sun is obviously rising from the left, that is, from the West.