Factual error: Throughout the episode the military are shown carrying Israeli Military Industries Tavor assault rifles. Israel does not sell weapons to any Arab states. Especially if said state already allied with the US and is thus buying its weaponry from the US.
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.
We Are Never Ever Getting Back Together - S2-E3
Continuity mistake: When David Wilson is talking to his son in the garage, his head is turned towards his son, then in one shot it's instantly tilted down, then instantly back towards his son in the next shot. And after the shot goes back to where his head is turned, he says "sorry" without his mouth moving.
Continuity mistake: The scar on Singleton's face changes from the right side of his face to the left side several times.
Character mistake: At the end of the scene in the grocery store when Rachel collapses and people run to her aid, the camera pans out to reveal a sign that prominently reveals a sale on "Pop Tards" or possibly "Pop Taros."
Continuity mistake: The girl in the flooding tub is weighted down and has her hands tied in front of her. When the police lift her out, the weights are gone and her hands fall to her sides, and then are tied again when she is on the floor.
Factual error: The only person ever addressed as "Sir" is the brigadier, and then only occasionally. No other officers ever seem to be addressed in this way.
Factual error: Congressman Tate and his brother are on the shooting range. They are wearing high-tech electronic earmuffs that block harmful noise like gunfire while amplifying speech and other low-volume sounds. Although, they are wearing high-tech earmuffs, they both have an ear exposed so they can hear each other talk. (00:13:00)
Episode One - S1-E1
Factual error: In the fairground chase the characters jump over some Heras fencing that wasn't invented until the next decade.
Factual error: Helmuth von Moltke is addressed and referred to several times as Field Marshal. He wasn't promoted to that rank until 1871. He was a lieutenant-general during the Second Schleswig War in 1864.
Episode #1.3 - S1-E3
Other mistake: The publican asks for payment for some drinks, saying "One and six, please", meaning one shilling and sixpence. The subtitles say "One-o-six, please."
You Always Hurt the One You Love - S1-E1
Factual error: In the opening sequence, Dr Frank Winter is hitting golf balls in the desert while a radio announcer is reading off names of Americans killed in action. One name is identified by the rank of "Lance Corporal," which the US Marine Corps first adopted in 1958, long after the war. Several minutes later, Dr Winter dresses down an Army MP, referring to him as "Sergeant First Class." During WW2, that rank was actually "Technical Sergeant."
Character mistake: At Houdini's performance for the Kaiser, a German officer with the rank of Lieutenant Colonel (Oberstleutnant) is asked to assist. Houdini at first correctly addresses him as Lieutenant Colonel, but then incorrectly refers to him as Lieutenant several times after that.
Factual error: Constantine's driving licence is shown - it's a UK licence, but no. 7 where his signature should be is blank, and no. 8 where his full address should be simply says "England, United Kingdom." Both are printed onto the licence when it's issued - without that info it would never be issued in the first place. And it's not a fake, because why would he have a fake licence with his real information on it? It's just an incomplete prop.