Plot hole: Carter is apparently just learning about Feynman's Day, which Allison explains as "Eureka's version of April Fool's Day." It's highly improbable that Carter could have lived in Eureka for five years and still be oblivious of this, especially given all of the hijinks that happen.
Plot hole: If we assume that Dexter took off the oil cap while it was still in the shop for Arthur, it still doesn't explain how Dexter got the jump on him when his car (hours later) broke down. Arthur's not shown to stop, but to merely drive for hours on end until night. Arthur's on a barren strip of road with no other cars or traffic to be seen for miles. It cuts from Arthur looking in his engine to Dexter suddenly walking out from behind the Mustang. The only explanation is that Dexter waited in the car with Arthur, which doesn't make very much sense. All this takes place in the last 15 minutes of the finale.
Plot hole: In "Into Hell, Part 1", season 4, episode 7, the Unit discovers a soldier who they think is Betsy, tied to a chair. They note that the room is covered with PIRs or Passive Infrared Sensors. They then take out their night-vision goggles and begin counting the number of "infrared lines" they see. However, PIR sensors don't emit anything; they are passive. (00:23:15)
Plot hole: In the Season 2 episode "The Homecoming", when Mr. Ballentine is telling Emily about her assignment, he sets the blue folder down near his side of the desk before he gets up. Then we see Emily pull it across the table and quickly peek in it. When Ballentine turns around, the folder is still close to Emily, which shows that she at least tried to look in it, yet Ballentine says nothing about it no longer being in the spot where he set it down. If Emily thought that it was okay for her to look in the folder, she wouldn't have waited for Ballentine to turn around before she touched it.
Plot hole: The flashback involves Young Shawn and Gus getting ready to go to a comic book convention, Shawn in costume, with his father's consent. This contradicts the flashback in "Shawn Vs The Red Phantom," where Henry won't allow Young Shawn to even read comic books or play superhero with a towel around his neck.