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 number of times throughout the season they refer to LSD being found in dead bodies and LSD being matched with other LSD found in other (living or dead) bodies. LSD starts a chain reaction in the brain which requires about an hour to take a noticeable effect. And, by that time, it has completely broken down and no longer exists as LSD in the body - before the person even begins to get high. There is no test for LSD even in the most bombed living individual, let alone a dead body years later.
Continuity mistake: In the overhead shot of Rust and Marty driving to the Dora Lange crime scene, the cornfields show no signs of being burned. Based on the amount of fire shown in the episode's opening scene, that should be very visible.