Continuity mistake: Just after Vesper has shocked James, the lead connecting the defibrillator to the chest pad can be seen to have come loose. When the camera cuts back to James, the lead has been reconnected. Note this is after the shock, not the disconnected lead that forms the drama.
Continuity mistake: When Bond is being tortured in the seatless wicker chair, in one shot when you look at the right side of Bond's face the blood is running in an unbroken line down his face. The next shot where you see his face the blood has been smudged, but in the next shot, where you see his right cheek, the blood is running unbroken again. This happens a few times in this scene.
Continuity mistake: At the Poker game between Bond and Dimitrios the chips change place during shots. Most noticeable on Dimitrios' small stack (two black chip stacks and one green) in the overhead shots they are on the black line of the table carpet, in the shots from the sides, they are in between two black lines. The same is with Dimitrios' cards and his good luck token.
Factual error: When James Bond is swimming underwater, trying desperately to save the trapped Vesper, if you look, he is constantly yelling and exhaling, sending bubbles from his mouth. The problem is that this continues almost constantly for at least a full minute. Nobody has the sort of lung capacity that enables them to expel that much air and still function underwater for that long.
Continuity mistake: At the end of the scene where Bond rolls the Aston Martin, during the wide shot just as the car finishes the last roll, the windshield is still on the car. In fact you can see the windshield fall back into a "normal" position. However, when the shot changes to the close-up of Bond in the car the windshield is no longer there.
Continuity mistake: When Carlos the bomber goes through the security door at Miami airport, he enters the door from a brightly lit area. We then cut to him emerging from the door - behind the door is completely dark. We then cut back to the door closing from the other side, showing that it's definitely intended to be the same door.
Visible crew/equipment: In the scene where Bond is following the bomber in Miami Airport, when the bomber goes through the security door using the ellipsis password, and proceeds down the corridor,Bond tries to catch the door before it shuts. There is a hand visible which pulls the door shut for a brief second and holds the door in place.