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 the scene where Bond is chasing Silva in the Underground, he leaps over the emergency buttons and slides down the escalator panel. Initially you can see the emergency buttons at the bottom of the escalator, which would have broken his slide, however, these "disappear" as Bond slides smoothly off the panel at the bottom.
Continuity mistake: In the scene where M (Judi Dench) is been driven back to the MI6 building, her car is stopped on a bridge and you see two cars stopping behind M's vehicle while she is still inside. She then gets out of the vehicle to speak to a police officer who requested that the car stop, and now two different vehicles are behind.
Factual error: James Bond is seen running down Whitehall in London. Behind him a number 38 bus is seen. However the 38 does not travel down or indeed particularly near Whitehall. This was probably done to show off the new Routemaster bus, which is currently only used on the 38 route.
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.
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: As Bond chases Patrice over the rooftops of the Grand Bazaar in Istanbul at the start of the film, in the close ups of Daniel Craig his tie can be seen noticeably flapping over his shoulder (as you would expect). But in the long shots with the stuntman it doesn't move about at all, presumably having been fixed in place for safety.
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).