Character mistake: Luther and Ripley pretend they have not been on the boat and did not find the victim's body as they do not have a search warrant and do not want to compromise the case. However, under English law a police officer may legally enter any premises without a warrant in order to save life, prevent injury or protect property. Since they have a reasonable belief that the victim is being held prisoner on the boat and her life is in danger (as far as they know at that time she is still alive), they don't need a warrant. As experienced police officers, they are bound to know this.
Other mistake: Ian is pointing a gun at Luther in the hotel. The maid starts to open the door from an adjoining room (roughly 4 meters away). Ian starts firing at the door, and Luther falls over on the spot he is standing on. Half a second later, Luther is through the door pushing the maid to the floor. What about the hail of bullets he must have had to go through to get to the door? (00:47:00)
Factual error: A uniformed constable addresses Detective Sergeant Ripley as "Guv" in this episode and another officer addresses him as "Sir" in the following episode. These terms of address are reserved for officers of the rank of Inspector or above. Sergeants are addressed as "Sarge", "Skipper" or "Skip."
Factual error: The detectives determine that the killer must be using a boat which he used to dump the first body next to a canal in Birmingham. However, the barge he is using is far too large to navigate the Grand Union Canal from London to Birmingham, let alone the canals within Birmingham itself.
Other mistake: During the opening sequence. Sadie Buckingham walks out of Smithfield market, walks down St John's Street, Cawcross Street, then up Greenhouse Rents. This is a dead end. She next appears walking along Jerome street where she is murdered. After the titles DCI Luther walks down through the police cordon and meets Martin. In the background is Godfrey Phillips Ltd, which is a company at the junction of Commercial Street and Jerome Street. This is a 30 minute walk from Smithfield. Luther then says "is he having a joke... slaughtered like cattle next to a meat market".
Other mistake: A detective is watching footage on a computer screen of the standoff with Luther and Reed where police snipers are on rooftops. Someone walks across the screen right behind Luther and Reed. Surely the police would have cordoned off the area with the snipers on the rooftops.