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.
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 Bond confronts the assassin and lets him fall to his death from the building in Shanghai, he then looks over to the opposite window where Severine is standing in the smashed window hole, her hair blowing in the breeze. When it cuts to the closeup of her leaving you can see the light reflected in the window glass that has magically reappeared.
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).
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.
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).