Episode #1.3 - S1-E3
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.
It Happened 'That Night' - S3-E1
Character mistake: Emily's mom tells Emily that reporters were there to talk to them about "Emily's missing remains." It's supposed to be "Allison's missing remains." (00:17:00)
Character mistake: In the opening scene, Sherlock makes a point to correct Barry's poor grammar. But later, when talking to Watson, Sherlock uses poor grammar by saying "who's sleeping with who." It should be "who's sleeping with whom." Not a mistake Sherlock would make.
Character mistake: Nathan suggests to Dwight and Audrey that maybe he should shut down the meeting of troubled people. As chief, he should know he has no authority to stop citizens from assembling.