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.
Factual error: In Turkey the first two numbers of a vehicle licence plate belong to a city. Number 34, for example, is for Istanbul. The motorcycle chase in the beginning of the movie takes part in Istanbul (both the Land Rover and the Audi have "34" plates). However, once they come to the bridge where Bond jumps to the train, all the vehicles on the bridge (except the Land Rover) have plates starting with "01", which is the licence plate number for the city of Adana, where those scenes were shot.
Factual error: On the District line, Temple and Embankment stations do not have two platforms between each train. They actually run side by side, and not with the platforms in between the two trains.
Factual error: When Bond climbs up from under lift/elevator after hanging under the assassin's lift car he opens adjacent lift/elevator landing doors just by pulling them apart. This is not possible without first disengaging the door lock mechanism, which is operated when car is at that level. If this were possible anyone could push lift/elevator doors open from any level and fall down the shaft.
Factual error: It would be impossible to drive a silver Aston Martin DB5 from London to Scotland without getting splash marks on the side of the vehicle. Especially in view of the fact that the road conditions are wet by the time 007 and M reach the Scottish Highlands, but the car still looks immaculate.
Continuity mistake: Just after Bond has lost track of Silva in the London Underground station, the scene cuts to M's parliament meeting. At this time, the time on M's watch reads 3:00. Yet when Bond catches up with Silva in the basement of the London Underground station, the time on Bond's watch reads 4:55. Far too long to have passed given the events seen and distances travelled.
Other mistake: In the opening scene where Moneypenny is holding the rifle, you can see the flash suppressor was installed upside down. The 'slots' are on the bottom.
Continuity mistake: In the courtroom shoot out scene, Bond shoots two fire extinguishers to use as a smokescreen. If you watch Eve as he steps out, she lowers her weapon dramatically at the last second. When the camera cuts to her immediately after, her gun is raised again.
Factual error: The Ambulances used in the film in the "M" attack in London are incorrect, as they are not the current vehicles used by London ambulance service. They all now have the patient lifts on the rear. Various vehicle tax discs were out of date too.
Factual error: When Bond is chasing Silva through the London Underground system they both supposedly get on a train at Temple Station on the District Line. The train is in fact a Jubilee Line train, but with District Line route maps on the inside. The back said Elephant & Castle and the front said Wimbledon.
Continuity mistake: While talking with Severine in the casino bar, Bond gets his martini in a frozen glass. He picks it up, the camera's point of view changes, and the glass is instantly completely unfrozen.
Continuity mistake: In the scene in the National Gallery early on in the film where Bond meets Q, there is a line of bench seats. Bond is sitting on the end of one benches looking at a painting. The shot is from behind. Q comes and sits on the adjacent bench next to Bond. You can see the gap between the two benches between the pair. Then the shot changes to a front view as they start talking. But suddenly there is no gap in the bench in between them, they are both sitting on the same bench, the gap is now to Bond's left.
Factual error: In the car scene just before the MI6 explosion, invalid IPv4 addresses (with digits larger than 255) are displayed on the laptop screen. [This gets debated, with comparisons made to 555 phone numbers. 555 may not be a real area code, but there's nothing technically preventing it being one. IP addresses, however, just can't have numbers larger than 255 - it's like a phone number having a % symbol in it.]
Continuity mistake: When Bond drives his motorbike into the bridge, the scene changes to Bond landing on the train, the bike magically landing on its wheels upright, then when the scene changes again, the bike has disappeared, yet the bike should be seen rolling out of control.
Continuity mistake: In Istanbul, Bond goes to the mosque. When he approaches the mosque, the exterior shot is of the Sultan Ahmed mosque or "Blue Mosque" (http://en.wikipedia.org/wiki/Sultan_Ahmed_Mosque). When Bond is inside, the interior is actually of the much older, nearby "Hagia Sophia" (http://en.wikipedia.org/wiki/Hagia_Sophia).