Continuity mistake: When M arrives to see Mallory, a bag is placed on the ground near her seat. It is still there just as she stands up and grabs her coat. In the next shot, the bag is gone and she walks out without it.
Revealing mistake: When the train plummets into the basement, the windows never seem to smash nor crack, nor does the paint on the train ever seem to scratch the slightest bit. (01:34:15)
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)
Deliberate mistake: In the chase through the tube station, Bond and Silva both slide down between the escalators. On the London Underground, this area has hard vertical signs at regular intervals, precisely to prevent people sliding down. (01:31:50)
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: 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.
Continuity mistake: When Silva is first seen in the London Underground station, he walks down a flight of stairs. On the left side of the stairs, there are two men. In the next shot, it is now only one, an entirely different man. (01:28:00)
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: When Silva is about to slide down the escalator, Bond runs past the same blonde woman twice between shots. (01:31:40)
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 courthouse despite having no way of knowing that she would be there that day.
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.
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.
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)
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. (00:57:15)
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)
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. (01:37:45)
Suggested correction: His plan was always to blow a hole in the tube to have a train crash at that location. This would have caused mayhem and take up a lot of resources (police/ambulance/fire). Bond chasing him had no impact on this part of the plan. Then when his team picked him up, his team would have known where M would have been on that day and would have driven him there. With resources on high at the train crash site, it would have made it easier to reach M.
XIII
This aspect is a bit of a plot hole for me too - one of the factors outside Silva's control is when he can escape MI6 custody, which depends on when Q triggers the 'Granborough' Trojan, which Q only spots with a bit of help from Bond. That triggering could have been hours, days, weeks or never - pretty fraught with risk to rely on individual(s) making that happen at a plot-convenient moment... otherwise Silva has a bunch of associates hanging around London for a few days permanently on a few minutes alert for a random event.