Plot hole: When Eve is being led out to the airplane, she bolts when two shots are fired from inside the house (presumably Anna firing at Thornhill, not realizing that the gun was loaded with blanks). Moments later, she jumps into a car driven by Thornhill, and they escape together. There is no way Thornhill could have gotten there that fast, given that he was in the house only seconds earlier.
Plot hole: When the flying saucer first appears, it is supposed to be 4 AM. But the grave diggers are still there from the burial of Bela Lugosi's wife.
Plot hole: The radar operator reports a bogey at 15,000 yards, without, at first, giving a bearing, yet everyone on the bridge immediately looks to the starboard bow.