Continuity mistake: When Bauer walks up the stairs in front of Waterloo station, the clock on the building is displaying 1:50pm, however we are only about 8 minutes into the episode. (00:07:50)
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.
The Hong Kong Kowloon Treasure Murder Case File 1 - S1-E1
Other mistake: Throughout the episode, certain street signs ("Job&Piece", "Ocean Guest House") are used multiple times in different points of Hong Kong (for instance the establishing shot of the city, behind Kindaichi and his friends when they meet, during the bike chase).
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.
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."
Plot hole: After Jack's wife disappears for a couple of days, he receives a call from her cell phone from the cab driver who found it. Jack arranges to pick it up from a hotel. After he picks up the phone, its battery dies as he checks it, at 12:13pm. In the next scene with Jack, he's in a bar with the phone plugged into a charger; when it comes on, he checks it - the battery shows 3%, but the time is 9:13pm. He apparently wasted 9 hours when he's in pursuit of his wife's mysterious disappearance before plugging in the phone. A scene later, the time is now just a minute later at 9:14pm but the battery is now charged to 79%. (00:37:25 - 00:40:45)