Continuity mistake: In the breakfast scene, Nightcrawler teleports onto the chandelier to grab food. However, in previous shots we've seen of the dining room, there was no chandelier anywhere near the dining room table.
Plot hole: In the end of the episode, Wolverine is seen recuperating after surgery removes the microchip in his brain, and his head is bandaged. Performing invasive surgery on Wolverine's brain isn't possible, since that would require going through his adamantium-infused skull and no conventional material is capable of that. Only a less invasive technique (such as going through the cranial sutres or cranial foramen) would have been effective and that most likely would not have required Wolverine's head to be bandaged (especially with his rapid healing mutant ability).
Continuity mistake: The claw marks Wolverine slashes through the door are suddenly longer when he exits the room.
Mutant Crush - S1-E4
Plot hole: With Jean's powerful telekinesis, she should have been able to untie herself from the chair.
Suggested correction: She would have to see the knots to untie them mentally. Blob could have used any number of ways to tie the rope off.
He didn't use rope, he bent a pipe around her so she should have been able to manipulate it.
Other mistake: The trees that the brainwashed Wolverine slices down, while chasing after Kitty, fall to the ground too slowly. They should fall quickly as a large, heavy object would, but instead float down like a feather.
Other mistake: The guitar next to Rogue's bed has position markers on all of the odd-numbered frets, beginning with the first. Position markers on guitars start on the 3rd fret, and sequentially follow on the 5th, 7th, 9th, 12th, 15th, 17th, 19th, and 21st frets (and 24th where applicable).
Strategy X - S1-E1
Other mistake: When Nightcrawler pumps his fist in elation while standing in front of the mirror, his arm casts no reflection. (00:09:52)
