Continuity mistake: When Magneto lands the Golden Gate Bridge on the island, as it hits land it is still very light. When the Brotherhood start to walk toward the island, it cuts to a wide shot of them still walking on the bridge, and it's suddenly dark.
Factual error: Take a look at Google Earth - the section of the Golden Gate Bridge Magneto breaks off (tower to tower, with a bit of overhang, 0.9 miles at most, more like 0.85) is not long enough to reach Alcatraz - the two closest points are 0.96 miles apart, and that's the very end of a pier, which most likely couldn't support the weight of the bridge anyway.
Continuity mistake: When Logan pulls out the two horns from his stomach, the camera shows two holes on the singlet as part as his wounds. In the following scenes, the holes of the t-shirt have recovered (I know Logan has healing ability, but not his clothes).
Revealing mistake: In the scene in the laboratory under the mansion, when Wolverine confronts Jean about Cyclops, there is a close up shot of Wolverine's hand holding Cyclops' glasses. In this shot there is a small cut visible on Wolverine's hand - his healing power should not allow this.
Other mistake: When Warren/Angel refuses the cure and spreads his wings, he is looking to the left of the screen at his father. In the reflection of Dr. Kavita Rao's glasses the shot of him is not reversed; he is still looking to the left of the screen when he should in fact be looking to the right.
Continuity mistake: In the scene where Angel is about to be given the cure, there is a change in the hypodermic syringe used to administer the drug. As the camera shot switches between views of Angel being strapped in and Dr. Rao preparing the injection, the first view shows the hypodermic needle cover already having been removed. The camera switches to Angel then back. Dr. Rao, having completed the preparation for the injection, proceeds to remove the cover which had already been removed in the first shot.
Visible crew/equipment: When Wolverine grabs Cyclops's sunglasses at Alkali Lake, a crew member, walking by, is reflected in the lens.
Other mistake: At the beginning of the credits, (of the version shown in cinemas) they have mis- spelled 'Starring', they put 'Strarring'
Plot hole: When Magneto lands the bridge between San Francisco and Alcatraz, only the ends of the bridge are on land, with the rest still suspended in the air despite having nothing to keep it there. Magneto could arguably be using his powers to keep it supported, but at the moment he loses his powers it still does not collapse at the center.
Continuity mistake: When Wolverine approaches Xavier's grave, the shadow of his head falls onto the grave. Then, after Jean has stopped telepathically speaking to him, the shadow of his head is left of the grave, even though Wolverine is in the same position. There was not enough time for his shadow to have moved that much.
Continuity mistake: In the scene on Alcatraz, Collosus throws Wolverine at Magneto. After Wolverine is dragged along the floor and Magneto says something to him, Wolverine looks up at Magneto from the wrong direction (this is explained when you look at the deleted scenes but doesnt fit in the final cut).
Continuity mistake: When Angel is about to get the "cure" in the San Francisco skyscraper, you can see the reflection of San Francisco in the outside of the building - a huge hill with lots of buildings, and a large white tower at the very top. But when the camera switches to inside the building, looking outside, the view has suddenly changed to a view of the bay, and Alcatraz.
Continuity mistake: When Magneto and Charles Xavier are in Jean Grey's house (when she is older and making everything around her float in the air), watch the book shelf when it falls down. At least 25 books fall out of the top shelf and in the next few shots you see that the number of books fallen has minimized greatly.