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 they're sitting on. From behind, Bond's coat tail is barely covering the gap between two of the benches. From the front though, his coat tail is further back, covering the gap almost entirely. (00:37:05)
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. (00:45:50 - 00:47:20)
Factual error: In the opening scene on the train, the coupling is shot away and the rear wagon separates. This ruptures the continuous brake pipe, which would cause both halves of the train to stop immediately.
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. (01:31:50)
Character mistake: There is no way an operative of Q's position and experience would have plugged a device taken from a bad guy into their network. Especially as they had had a security hack only a few days earlier and this was a security establishment. It would have been thoroughly tested and checked on a stand-alone, air-gapped setup first.
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. (00:10:35)
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. (00:20:15)
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, and destination 'Wimbledon' on the back. It was filmed in Charing Cross (with Circle/ District line signage added) and the platform scene was filmed on the disused Jubilee line platform, from before that line was extended. (01:29:40)
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. (01:36:25)
Plot hole: Silva spent years planning his revenge against M, yet the ability to carry out his plan depended on conditions that were entirely out of his control, such as being captured by MI-6 despite having no way of knowing that Bond was coming for him, and only escaping custody because Q triggers the trojan - there's no way Silva could predict when that would happen, could have been hours, days, weeks, or never. He then ambushes M at the court house despite having no way of knowing that she would be there that day.