The price of the X-Men comic book was 40 cents, placing its publication to around 1979. But the comic books in the movie starred the "real-life" X-Men, including Wolverine, who didn't come into the public scene till 2000.
During the opening scene where the men are attempting to steal the wheels from the limousine, we can hear the distinctive sound of an air-driven impact driver, as used in commercial tire shops. When we see the outside shot of the man actually removing the lug nuts, he is seen using a standard 4-way lug wrench, and no air tools or air lines are anywhere to be seen.
Logan's entire skeleton is made of adamantium, the hardest metal in the known universe which can only be damaged on a molecular level. Despite Logan's failing healing factor, he still could not have been impaled in the way that he was, due to his adamantium skeleton.
When Xavier tells Laura that Shane (the Western they're watching on the hotel TV) is nearly a hundred years that is not quite true. As that movie came out in 1953, it would only be seventy-six-years old in 2029. Hence, the phrase "...almost eighty years old" would be more accurate.
Logan and Charles are in the limo going to pick up Laura who is fighting Pierce's men and the police. As Logan is steering the car, there is a very brief shot of the backseat, and Laura is already sitting there.
When Pierce enters Logan's Chrysler, there was a lady visible through the windshield walking towards Logan's car. The scene focusses on Pierce in the backseat and after some time focusses on Logan in the driver's seat again. The same lady was seen at the same place at the same direction.
An adamantium bullet was shown to be able to cause significant damage to adamantium by ripping off half of X-24's skull while Wolverine and X-24 are unable to damage each other's skeletons with their own claws.
When Laura gets into the vehicle to drive while Logan was resting, she doesn't bother to adjust the seat. Considering the difference in height between her and Logan, there's no way they'd both be able to drive in the same seat position. Even sitting on the edge, her feet would barely touch the pedals.
The video on Gabrielle's mobile is quite nicely edited and narrated for supposedly being a series of underhand recordings. The voice over even extends over several scenes. Being on the run, Gabrielle would hardly have time to do this - even if she had the skill, equipment and software to do it.
Caliban likens his weakness to sunlight to the character of Nosferatu. However, in the film Nosferatu there is no character by that name; the vampire's name was actually "Count Orlok," with Nosferatu being his species. However this was intentional as the general audience will not know who "Count Orlock" is but will know that Nosferatu has to relate with Vampires. This would explain why Caliban burns when exposed to sunlight like a Vampire which Nosferatu also means.
In the fight scene at the end of the film during one shot of Logan running, his metal claws jerk up and down slightly out of place as his feet hit the ground, revealing that they are actually special effects appliances held in Hugh Jackman's hands and not fixed blades that extend from the character's skeleton.
When the main characters are attacked in the casino hotel in Oklahoma City, Logan uses the elevator to return to their room to administer Prof. X's medicine. When Logan gets off the elevator the down button is clearly pressed but nobody is waiting in the hallway to get on.
The goof items below may give away important plot points.
When the kids are being chased in the woods by the Reavers, Laura can be heard shouting: "Corred!" ("Run!"). This is how it's said in Spain (from where Dafne Keen learned to speak it), but since her character was raised in Mexico, she should have said "Corran!".
At the final scene with Laura and Logan, one shot of Laura shows a tear rolling down her right cheek. The next shot her cheek is dry, and she has a dribble of snot from her nose. Then a third shot has the wet tear line back on her cheek and her nose is dry.
When Pierce comes to talk to Logan after returning from finding Gabriela murdered, Pierce reaches his mechanical hand forward and places it on Logan's shoulder. In the reflection of his sunglasses you can see his real hand wearing the green screen glove, at 28 minutes 47 seconds .
When Logan attacks the mercs who are trying to capture Charles Xavier, he clearly drives his claws through their brains. SInce Charles power over them is based his telepathy, the destruction of their brains (or even serious damage caused to them) should have caused them to collapse as there was nothing left for Charles to control.