Factual error: The Mountain Dew logo seen in the stadium didn't come out until 1980, despite the film being set in 1973. The changes made in 1980 were subtle, but still distinct, especially the loss of the "wave" in the "D" and the smaller hole on the "a."
Bishop73
Continuity mistake: Charles has just denied Logan's request for his help and is now in his bedroom. There's a photo of Raven on the right side of his bed with an ashtray slightly behind it. A couple of shots later, the ashtray is gone and the photo has moved closer to the bedside table's edge. (00:33:50)
ployp
Trivia: Len Wein, who created the character of Wolverine, has a cameo as one of the Senators that turn down Bolivar Trask's sentinel project.
wizard_of_gore
Trivia: Director Bryan Singer has a cameo as one of the photographers on hand when Mystique jumps out of the building after she attempts to kill Bolivar Trask.
wizard_of_gore 
Trivia: When Magneto gets his helmet back there are two more items in the room, the suit of Havok and one wing of Angel, both from X-Men: First Class. (01:31:55)
ozwal13
Continuity mistake: Logan has just woken up in the past. In the first shot of him, the blanket is nowhere to be seen. In the next shot, the blanket now covers his stomach. (00:17:40)
ployp
Chosen answer: Government agents were previously aware of mutants - they are probably the same ones that imprisoned Magneto, and his attempt to prevent JFK's assassination was most likely covered up due to the revelation that Kennedy was a mutant. Magneto was already in prison by the time Trask pitched his Sentinel idea to the government, which is probably why the program wasn't instituted.
Phaneron