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.
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 Bond is stalking Patrice before he kills the mark, 007 is not wearing gloves. Later, when he is seen holding Patrice before his fall, he is wearing black gloves.
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: Both tube stations feature yellow walls and signs of the Circle line, but Bond is travelling on the district line, meaning they should be green (or green and yellow).
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.
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.
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 to 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.
Factual error: During Bond's fight with Patrice in the Shanghai high-rise, Patrice's sniper rifle fires several times, presumably by accident. However, moments earlier, when Patrice assassinates his target in the neighboring building, the rifle is shown to be bolt-action. A bolt-action rifle can't be fired more than once without manually re-cocking the gun in order to fire the next round.
Continuity mistake: In a hotel room at the start, Bond finds Ronson bleeding. Note the amount of blood on his chin coming from his mouth. In the shot where Bond opens Ronsons's jacket partway to see the blood, there are a bit of dripping blood going down his chin. In the shot where Bond leaves the room, there is suddenly a huge blob of blood on the left side of the Ronson's lips that wasn't there before.
Deliberate mistake: When Bond walks through the tube train he leaves the doors open so the cameraman can follow him. This scene was shot several times to get the front and rear shots. When filming from the front, he closes the doors behind him.
Visible crew/equipment: After Patrice jumps onto the train, Bond drives off the bridge and lands on the train and begins to fall over the side, then climbs back on. Eve then says "They're on the train." During this shot, you can see the white platform Bond was standing on to climb back onto the train.