Continuity mistake: When the Munchkins come out onto the Yellow Brick Road, one of them comes out of a sewer hatch on the yellow bricks. It's located on the ground in front of the first small white cottage with the thatched roof, right behind Dorothy's house. Throughout this scene, the dark hatch is visible whenever we see that specific location, such as when the Wicked Witch of the West asks, "Who killed my sister? Who killed the Witch of the East? Was it you?" but as Dorothy begins to "follow the yellow brick road", there is an overhead shot followed by a front shot and the dark hatch has disappeared.
Visible crew/equipment: Near the very beginning, when Dorothy is with Prof. Marvel, after she runs out of his trailer and the wind starts blowing, he follows her and goes towards his horse tied up outside. As he's walking towards the horse, you can see the shadow of a boom mic along his trailer. (00:14:55)
Revealing mistake: After the witch says "Hey scarecrow, wanna play ball" she throws a fireball at him. But you can see the ground catch on fire before the fireball hits the ground. (00:46:20)
Continuity mistake: There are two actresses playing Dorothy (and two dogs playing Toto) as she opens the door to Oz. The first is a stand-in wearing a dull colored dress in order to make it look black and white. After she opens the door and moves out of the way, Judy Garland is who we actually see walking through. Look at the different pleats on the two dresses. (00:19:25)
Continuity mistake: You see a Flying Monkey with Toto before he gets picked up. (01:15:30)