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.
Episode #6.4 - S6-E4
Character mistake: The character pretending to be Lt. Ellis refers to himself as First Lieutenant. That rank is referred to orally as "Lieutenant", not "First Lieutenant."
Reika Hayami and the Uninvited Guest - S1-E5
Continuity mistake: When Reika asks everyone who got the first two questions right to stand, the imposter's cup and Kindaichi's are different from just a moment before; Kindaichi's is suddenly empty and both cups are turned (the guy's cup and plate switched side too). (00:10:20)
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)
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.