Continuity mistake: As Banacek approaches the room in the farmhouse that has the cross, he has a cigar in his mouth. As he enters, the cigar is gone.
Factual error: Noticed this watching season 2, and I assume it was also the case in season 1. The entrance of the Miramar Playa has flags from various countries flying. One of them is the Canadian flag with the maple leaf. Wrong flag. In that period the Canadian flag was the "Red Ensign", or "Dominion Flag." The maple leaf flag wasn't introduced until 1965.
Death by Resurrection - S1-E12
Continuity mistake: McBride and Ryan are driving through town in heavy traffic. When the shot cuts (and their conversation continues uninterrupted), not only does the rear view mirror disappear, but an obvious rear screen projection changes their locale to a country road with trees and no traffic at all. (00:06:00)
Factual error: About 45 minutes into the episode, the girls' car stalls because the alternator's wire has been removed to cause this. For a car to stall like this, the battery has to be heavily drained which it would wind up being. When the killer comes up to the car, he tells her to try and start the car. She does and it turns over but doesn't start. The battery would never have enough amperage to turn over the car if it didn't have enough to keep the motor running.
Triumvirate of Terror! - S3-E8
Continuity mistake: When the camera first shows Jimmy, he says "Where the heck have you been" and looks up to his left then straight forward. The shot changes to a close up and suddenly he is looking up to his right. (00:00:15)
Continuity mistake: The assassin is running across the square with the poisonous needle in his left hand in several shots. But moments later, when tackled to the ground, it is pressured from his closed right hand.
The Dark and Bloody Ground - S1-E3
Continuity mistake: At lunch, Jim asks Beth Davenport to find out everything Ann knows about her husband. She is touching her necklace, but at the cut that same hand holds a mug. (00:09:10)
This Must Be the Alamo - S4-E5
Continuity mistake: When McCloud and Broadhurst are driving around in the squad car, the vehicle changes in their closeups. Note the light atop the hood, the stickers with the police and unit number on the hood, the rearview mirror's vanishing act, etc.
The Secret Fate of All Life - S1-E5
Continuity mistake: The first time we see Detective Cohle being interviewed by the other two detectives, two of the beer cans on the table in front of him completely vanish between shots when he asks to see the rest of the discovery file.
Factual error: A 1973 Cadillac ambulance on the street is shown. The show is set in the 1960's.
The Legend of Curtis Ballard - S2-E3
Factual error: In the first flashback scene of Ballard serving a Mormon mission in Lithuania, he laments that he has been on his mission for 18 months and has yet to baptize a single person. In a later flashback scene when he and his companion are visiting the prostitutes, he states that he and his companion are 19 years old and they should be in college, going to parties, getting drunk, getting laid, etc. Instead of serving missions. These scenes take place in 1991, at which time young men did not serve missions for the Mormon church until after they turned 19, so if Ballard had already been on his mission for 18 months, then he should have been at least 20 years old, and closely approaching 21.
Other mistake: When Cobb starts to chase the red Dodge, there's a shot of him passing between two vehicles (and very distinct buildings in the background). When Murphy picks up the chase, this same shot is reused, showing her passing between the same two vehicles.
The Two Deaths of Jamie B. - S2-E4
Revealing mistake: When Naomi shows Kirsten the money, you can tell it's fake. Not only are the serial numbers all the same, you can read "this note is not real it is to be used for motion pictures."
Out of Time - S1-E3
Continuity mistake: When John meets Brooke and is examining her wrist, he ends up holding her hand palm up, but in the next shot her palm is facing down.
Plot hole: When Nicky goes into prison in Mexico to give a criminal a shot of insulin, two errors are seen. 1) The needle and syringe are far too big for an insulin shot, which takes a one ml syringe and a small sub-cutaneous needle. 2) When we first see the syringe, there is a small amount of insulin in it, but it grows the more times we see it, until a large syringe is two-thirds full - insulin is enough in a one ml syringe.
Borealis 301 - S1-E5
Continuity mistake: Near the end, Tom takes out the cipher key that was previously written on. But the cipher key used in the close up shots is different than the original. The words are spaced out and crossed out differently.
The Blessing of Many Fractures - S1-E8
Continuity mistake: During the fight with Zhou Cheng, Cheng does a forward flip and kicks Danny, making him stumble backwards. In the shot of Cheng lying on the ground, Danny is holding his stomach. In the shot immediately afterwards, he's holding his chest.
Ashes, Ashes - S1-E6
Revealing mistake: It's pretty obvious that Scott Glenn, the actor playing Stick, just has his hand wrapped up to look like it has been cut off. Both his hands are at equal length when his arms are down by his sides.
Continuity mistake: When Buzz and Provenza race out of the squad room to follow Rusty in the frantic search for Stroh, Buzz is in shirt sleeves with no tie. However when they arrive at their destination Buzz is wearing a coat and tie.