Audio problem: After Rogue was stabbed through the chest by Logan, she is sitting on the bench outside. Bobby/Mystique walks up to her from behind. Rogue says "Bobby" but her lips say "Hi". (00:42:25)
Continuity mistake: The chain Mystique wraps around Wolverine's wrists during their fight scene near the end of the movie mysteriously disappears shortly after that moment. (01:15:07)
Revealing mistake: When Magneto is lifting up Wolverine on the train, you can see screws on the bottom of his claws. His claws are natural, with metal directly bonded to them, and did not need to otherwise be attached.
Continuity mistake: When Logan is in bed in Xavier's mansion, Rogue walks in and tries to wake him. When Jean says, "Scott, grab a pillow," Rogue is still facing in Logan's direction, then it cuts and she is now facing Storm. (00:38:40)
Continuity mistake: During the final scene with Xavier and Magneto playing chess, there is a pawn missing from the board after Xavier leaves and Magneto pushes over his King. (01:33:47)
Continuity mistake: In the Statue of Liberty, Cyclops reaches up to his visor with his left hand and blasts away the door that was trapping him. Yet from behind he lifting up his right hand. (01:14:25)
Continuity mistake: In the Statue of Liberty, when Toad has shot his slime at Jean Grey's face, we see Cyclops shoot his optic beam to free himself from the room he was trapped in. Then he rushes to Jean's aid. If you look closely you can see that the gloves he was wearing, as part of his costume, are no longer there when he takes the slime off of Jean's face. (01:15:37)
Continuity mistake: When the police rush into the Statue of Liberty, a cop stands over a Statue Museum guard. In one shot the guard is on the floor with his left arm outstretched, but in the following overhead shot the arm is underneath him. (01:25:05)
Continuity mistake: In the medical bay at the end, Jean's medical coat keeps jumping from positions on her shirt between shots.
Visible crew/equipment: After Senator Kelly's death, watch Storm as she runs away. On her right, a microphone wire is visible. (01:00:40)
Suggested correction: Doesn't appear to be a wire, but rather a line in the fabric of the shirt she is wearing. There is one on the other side that you can see when she walks in to the room. Also, I don't think they wear microphones. Usually there is a boom mic overhead. You can see examples of the boom being used in the special features disc of X-Men 1.5 that shows you behind the scenes footage.
It is a battery pack for a mic. Bryan Singer admits this in the director's commentary.
Continuity mistake: Logan stabs Rogue through her chest, but there are no marks in the front of her dress.
Continuity mistake: After the last fight Logan is lying at the top of the statue with some wounds on his forehead. In the next shot those wounds increase size and have more blood on them.