Continuity mistake: When Charlie Allnut grabs a bottle of gin from the box, all the bottles in it are wrapped with paper for transport. But when Rose Sayer takes the bottles out of the box to empty them into the river, the paper is gone (from the movements of her hand/wrist, you can tell that she doesn't unwrap the paper from the bottles).
Continuity mistake: Alice Tripp is wearing different shoes when she starts walking home from the movie with George Eastman than she is when they are close to where she lives.
Continuity mistake: When the giant is fighting the bull, in the long shots the bulls horns are white, but in the closeups, they're black.
Continuity mistake: During the final dance number, Vera-Ellen's skirt alternates repeatedly between long and flowing and short and tight - perhaps to facilitate some of the dance moves.
Continuity mistake: Near the end of the film, Karl Malden is embracing Vivien Leigh. In one shot she has her left forearm against his chest, with her hand in the vicinity of his shoulder. In the next shot this arm is instantly around his neck.
Continuity mistake: When Jerry gets in Milo's car so that she can get cash to pay him for his paintings, they get into a dark green car. In the next shot they get out of a light (slightly greenish) blue car. (00:23:00)
Continuity mistake: After the first training montage, Alice is worried she might have flunked Meteorology and asks Marcie a question. When the door opens and everyone laughs, they both are in wildly different position than in the previous shot. (00:09:05)