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: The villains try to kill Cary Grant by getting him drunk and sending him off a cliff in a white Mercedes. He manages to get back on the road and they follow him in a Cadillac limousine. Cary passes a parked police car which gives chase. He brakes to avoid a bicyclist and the police car rear-ends the Mercedes. Immediately a blue 1941 Ford rear-ends the police car. Where did it come from? It would have to have been ahead of the Cadillac and very close to the police car to do so, but it doesn't appear until the moment of impact.