Continuity mistake: Bond and Silva are in a train with a number beginning in 96 and ending in 69 (96269, 96469 etc) However, near the end of this scene, the carriages have now become 96424 and 96224.
Audio problem: At the casino bar, Bond is talking to Severine. The camera angle switches behind Bond and he says "I'm guessing he was your way out" but his mouth doesn't move. The angle switches to facing Bond and you can see the side of his face move on his next line.
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.
Visible crew/equipment: In the Shanghai hotel, Patrice goes into a lift and Bond jumps onto the side of it. Whilst the lift is rising, Patrice looks outside of the lift. The shot cuts to an exterior shot of the lift. When it cuts the exterior shot, for just the first few frames a camera can be seen attached to the lift, used to film the shots of Bond.
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).