Character mistake: When everyone is preparing to go in to dinner, Robert is wearing an adjustable bow tie. It's bad enough he's wearing black tie to dinner when he has guests, as his mother points out, but Robert, the Earl of Grantham, absolutely would not be seen wearing an adjustable tie with a stand up shirt collar, even among only his immediate family. (00:35:15)
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.
Character mistake: Raylan Givens tells Judge Reardon that he carries a ".45 Glock" but this is untrue. Throughout the series with the exception of the beginning of the first episode Raylan Givens carries a Glock 17, which is only chambered in 9mm parabellum. His backup can be seen to be a Glock 26 which is also in 9mm.