Plot hole: When Bond climbs out of the lift he has been hanging onto underneath, how does he know the villain went right and to go that way? Yes, the lights come on automatically when someone walks past them, but Bond was climbing up and out of a lift. By the time he gets to the hall, all the lights are out.
Continuity mistake: When Silva gets out of the helicopter at Skyfall, the sky is approaching dark (7 o'clock to 8 o'clock sky). After throwing the grenades in the house and Silva says "Everyone listen to me! Don't you dare touch her, she's mine!" about 50 seconds later, it is suddenly pitch black (11 o'clock to midnight sky).
Factual error: When Patrice is going up the elevator in Shanghai, there is a shot of him taking the suppressor off his pistol. But once it's off you can see that the pistol barrel isn't threaded, which means that it couldn't take a suppressor in the first place.
Continuity mistake: When Bond runs out of the London Underground Station, he is covered in dust. The dust is still there when he runs into the Houses of Parliament and still there when he drives off with M. Yet the dust has disappeared when Bond and M find the Aston Martin. There was no time to clean up the dust since he was driving.