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.
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.
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.]
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.
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.
Continuity mistake: After Silva shoots Severine, he never puts his gun down. As Bond begins shooting the henchmen, Silva can briefly be seen, gun in hand, running backwards away from Bond. When Bond kills the last of the henchmen, and turns to Silva, Silva is unarmed.
Continuity mistake: During the yacht scene, Severine and Bond are standing looking out to the island. Severine is barefoot. Once on the island, she is wearing black heels. If she knew that she would need shoes to walk on the island, which is filled with rubble, she would already have them. She would have to walk back to her room on the yacht to get them. This is highly unlikely given the fact that they are being held hostage, and she could use her heels as a weapon.
Continuity mistake: During the pre-credits car chase Eve drives really fast and knocks a group of parked motorbikes that wobble, about to fall. From the immediate opposite angle the bikes remain undamaged and stand perfectly still.
Continuity mistake: During the first car chase right before Eve gets stuck in the traffic jam a shot inside the car shows that the right lane has barely no cars. Half a second later from an angle shot from outside the right lane is jammed.
Continuity mistake: While Eve drives and swerves to avoid cars, before she encounters the red truck two yellow taxis drive by and then a black car does. When the angle changes the second yellow taxi is driving by again.
Continuity mistake: During the motorbike chase, Eve gets stuck in a traffic jam and drives in reverse, damaging a yellow car's hood. In the next angle the hood is undamaged.
Revealing mistake: During the motorbike chase, after a man says "they're on the rooftop of the grand bazaar" Bond's bike jumps and right after there's a wide shot where the driver is not Daniel Craig but a stuntman.
Continuity mistake: Bond gets on the Tube at Temple. Then when he gets off at the next stop the first visible platform sign still shows Temple -before then changing to Embankment.
Continuity mistake: When M gets into the Jaguar making her escape from the court hearing she is in the rear offside seat behind Bond. A few seconds later, when viewed in the rear view mirror by Bond, she is on the other side of the seat, still belted up.
Visible crew/equipment: When Bond and another man are pulling into the "new digs", there is a shot of their black SUV turning right. In the reflection of the car, you can see a boom mic.
Continuity mistake: At the Shanghai Casino, Bond puts the very expensive chip on the bench between him and the cashier, who doesn't touch it, but leaves it behind as she gets her boss and the suitcase that holds a lot of money. As Bond walks away, it is gone.
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).
Continuity mistake: When Bond is on the train, before he gets into the backhoe, the number of people and cars on the street keep changing. Even if the train moved forward, the people/cars seen in the next shot weren't seen in the previous shots.
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.