Factual error: Everyone, including other police officers, addresses the detectives as "Detective" rather than using their ranks of "Inspector" or "Sergeant" as they actually would.
Factual error: DI Renton is a Metropolitan Police officer 'parachuted' into Kent to solve a big case. This hasn't happened for many decades. Local police forces investigate their own crimes without help from the Met.
Episode #2.3 - S2-E3
Character mistake: DS Maxwell comes to Katy's house to interview her but when he introduces himself he says "DI" Maxwell.
Episode #2.6 - S2-E6
Plot hole: Laura is spotted at an airport where it is assumed she goes to Edinburgh. Although she mentioned later on she broke her bail by leaving the 30-mile radius by her bail conditions. This wouldn't have been possible as part of her bail was to surrender her passport.