(at around 43 mins) Wolverine, Iceman, Rogue, an Pyro escape the mansion in Cyclops' Mazda RX-8, which is built with a Wankel (rotary) engine (which does not contain cylinders or pistons), but which sounds like a normal reciprocating engine (with cylinders and pistons) when Wolverine drives it.
(at around 13 mins) When Stryker is proposing the assault on the mansion, he introduces the surveillance photos as of a facility in "upstate New York;" by no stretch of the imagination is Westchester upstate. The town of Westchester is about an hour from Manhattan and Westchester County is right next to Bronx County, one of the five boroughs of NYC.
In Alkali Lake, as Nightcrawler and Storm are going to rescue the captured child mutants, when the camera pans to looking at the mutants with the back of the cell centre frame, you can clearly see Nightcrawler standing there before he actually teleports into the room a second or two later.
(at around 1h 6 mins) On the X Plane before the missiles hit it, Rogue scrambled to get her seat belt buckled. However, after the first plane flip, she's sitting with a smile on her face, hands in her lap, no worries. Then she's shown again gasping and holding the belt buckles with both her hands.
(at around 31 mins) Wolverine takes the soda from the cabinet and pops the top off. Before taking a drink, he hands it to Bobby (Iceman) to cool it off for him. In the next shot of Iceman, Wolverine's arm, and the soda bottle, the amount of soda has noticeably decreased. In the third shot, we see Wolverine and the now icy bottle with the soda level returned to the same level as the first shot.
When Wolverine fights Lady Deathstrike in the Alkali Lake lab, he slices through a metal pipe with his claws. One short piece of the pipe is cut out and falls to the floor. Since he has three claws, two pieces of pipe should fall to the floor.
(at around 1h 1 min) When dealing with the police, Wolverine gets shot in the head. It seems like the bullet penetrates his skull then is pushed out by his healing powers, however, the bullet only hits his Adamantium skull and sticks to his skull. Very similar to crashing a car into a brick wall and the car not bouncing back. Wolverine is knocked out by the force of having his head jerked at speeds of excess to 90 mph. His healing powers then push the bullet off his skull.
Since Senator Kelly died in the middle of X-Men, his subsequent appearances have been impersonations by Mystique. But any U.S. Senator's personal and political obligations occupy nearly every hour of each day. Given that there is only one Mystique, and that she impersonates several other characters in addition to appearing as herself, "Senator Kelly" must frequently disappear for hours or days at a time. In the real world or even the comic book world, her intermittent use of Kelly's likeness would be quickly uncovered as bogus by his family or staff.
(at around 18 mins) When Stryker visits Magneto to interrogate him, he enters Magneto's cell while still wearing his glasses, which have metal legs. The scanning process should have caught that on his way in; if it hadn't, Magneto should be able to manipulate them and use them to escape. It was shown in the first X-Men movie that Magneto can manipulate even non-ferrous metals, so he should have been able to affect Stryker's glasses.
(at around 17 mins) When Wolverine returns to the mansion at the beginning of the movie Jean tells him that she and Storm are headed to Boston, but the Professor hadn't tracked Nightcrawler there yet, so they shouldn't have known Boston was their destination.
(at around 18 mins) When Magneto is in his plastic prison he is reading a hardback copy of The Once And Future King. The book has small metallic corner protectors attached to it - something Magneto could have easily used to aid his escape.
(at around 1h 24 mins) Right before Wolverine leaves the assault team (Magneto, Mystique, Jean Grey and Nightcrawler) in the Spill Control Room, you can see the beard prosthetics adhesive base on his jawline.
The goof items below may give away important plot points.
(at around 2h) When the X-Men come to the President at the end, they hand him a blue folder. When they have left and time begins moving again, there is a green folder and random papers on his desk. The next time we see his desk there is only a blue folder neatly placed in front of him.
(at around 1h 55 mins) In the climax when Jean Grey is saying "I know... the only way", a shot is taken of Jean's face outside and the plane is visible in the background but none of the people inside are visible. Yet in the previous shot the X-Men are looking out through the glass.
(at around 1h 19 mins) When Mystique has impersonated Wolverine and entered the spillway doors in one shot she's almost near the doors when she shouts Stryker's name. In the next shot she's further away from the doors. In the next shot she's again by the door.
Mystique injects the guard a solution with high iron content so Magneto would be able to manipulate the iron in his bloodstream. The iron in this solution would have to be in the ionized state (valence of +2 or +3), but Magneto is shown to have power over metals but only in their unionized form (valence 0), which is the metallic state and not soluble in water. Even the metal detectors would not pick up the high iron content in the bloodstream.
(at around 1h 15 mins) In X-Men, Wolverine is able to spot Mystique twice by catching her scent. Once when she disguised herself as a small Statue of Liberty, and another time as Storm. Yet, he does not catch Mystique by scent when she masquerades as Jean Grey and tries to seduce him. Clearly his own feelings for Jean and this sudden arousal threw him off and kept him from sensing the impersonation.
(at around 1h 16 mins) At one point in the movie, Rogue, Ice Man and Pyro want to know where their X-Men uniforms are and Wolverine says they're on order. However, later when confronting the President, Rogue and Ice Man are wearing X-Men suits. However, this is one of Logan's frequent sarcastic answers towards juniors.
(at around 1h 45 mins) Jean claims that opening the door to Cerebro while Xavier is connected could kill him and everyone he's connected to. But that didn't happen when Magneto opened the door moments before, and Jean was connected to Xavier at that time, so she would know. However, Magneto had already used his power to shut down Cerebro before opening the doors, so Charles' life wouldn't have been in danger.
(at around 1h 55 mins) After Jean Gray sacrifices herself, there is a shot of the dam showing the breach and flooded area. While the entire foreground is flooded the water level is only a few feet below the top of the dam.
(at around 1h 55 mins) When all the X-Men and Xavier are talking with the president, supposedly everyone who is not an X-Man is frozen or time has stood still but in one shot you can clearly see someone blink a couple of times.