Continuity mistake: When Will tackles Andrew, his shirt is lifted up exposing a blue undershirt that's tucked in, which is also behind his gun. There's two shots of this, but in the next shot, right before Will punches Andrew, we don't see the undershirt.
Factual error: When Hap goes to the grocery store to buy beer, it shows signs of a grocery store selling beer, liquor, cigs, milk and bread. This doesn't exist in Texas. Liquor is not sold in grocery stores. And liquor stores (only sell alcohol, no groceries) would close at 9 pm and this is after that since everyone was asleep.
Episode #1.1 - S1-E1
Continuity mistake: The characters are looking at the Ten Little Soldiers. When one of them (playing the role of Blore) is banging on the glass, you can see the hands of the mystery writer ("Lombard") leaning against the base. He's leaning on the top of the display case in the next shot. (00:22:10)
Factual error: Huth does not wear his Knight's Cross correctly. It should be suspended below the collar, not attached high on the collar as his is.
Factual error: The DI refers to the local policeman as "Officer" Fuller. No British police officer would use this term to refer to another British police officer. She'd have said Constable Fuller or PC Fuller.