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.
While watching Shane Professor Xavier says he saw it when he was Laura's age. If true, then he would have been born in 1942. Since the movie is set in 2029, that makes him an octogenarian, not a nonagenarian like he corrected Pierce in Mexico.
In the initial fight scene, Logan gets shot with a shotgun. Although shot at with a gun, no additional wounds appear. Later, he is seen squeezing out bullets in the sink. Shotguns fire pellets, not bullets.
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 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.
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 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.
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 the actor'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.
Gabriela's phone, which Logan uses to learn of Laura's origin through video, has a Sony logo on it. However, the top portion of the phone's HUD (signal strength, WiFi symbol, battery percentage, and battery symbol) is that the iPhone, manufactured by Apple Inc.
The goof items below may give away important plot points.
In the Xavier's hotel seizure scene, when Logan struggles to get back to the room killing off the mercenaries standing in his way, the mercenaries should drop down the moment they are killed (at least those stabbed in their heads), since their minds can no longer be affected by Xavier's abilities. Instead they remain paralyzed until prof. Xavier receives his injection.