Plot hole: The protagonist gets in the exclusive, multi-millionaire, invites-only auction because the invitation is on a printed letter that he faked. This means that the guards at the entrance don't have a guest list to check, and since nobody knows who this person is, the staff does run a background check on his identity when he makes the first outrageous bid...by looking his name up on Wikipedia. That's mighty low standards of security, especially for an auction that was supposed to be for a selected audience and the most important in France.
Plot hole: Burke is murdered so that Doward will be sent out to replace him, as both hold the same rate of CPO sonar operator. But there are many CPO sonar operators in the Royal Navy. How can the GRU be so sure that Doward will be the man chosen to replace him?
Character mistake: Stabler calls for an LPR check on a vehicle. The description of the vehicle he gives is for white Mazda Sentra. Mazda doesn't make a Sentra. Nissan makes the Sentra. (00:09:45)
Continuity mistake: When Jake buys Chucky at the beginning of the episode, as it cuts back and forth between Jake's angle and the homeowner's angle, the way Jake is holding Chucky changes a few times. (Ex. In some shots, Jake is holding Chucky a few feet in front of him, in others he's holding Chucky right next to his chest, etc.)
Visible crew/equipment: When Jane straight up tells John Swift that she's taking a "vacation" so she'll be out of his hair, just as Jane walks out of her own office, the red T-mark is visible on the floor at the doorway. (00:15:45)
Episode #1.3 - S1-E3
Character mistake: Richie Gulliver is referred to as Rory Gulliver on the newspaper headline. As he's supposedly a very well-known Member of the Scottish Parliament, it's unlikely a newspaper would make a mistake like this.
Family Day - S2-E5
Continuity mistake: When Harry first picks up a water balloon and throws it, his hands are now empty. In the next shot, he has a blue water balloon in his hand. When he throws that one, you see it's a pink balloon being thrown.
Other mistake: Danny Stevens, an undercover police officer, is killed. The police agree to investigate the murder of his undercover persona, Steve Daniels, thus allowing the investigation of his death without compromising their ongoing investigation into the activities of a drug kingpin. The officer investigating the murder visits the kingpin to ask about Steve's murder. She shows him a photo of the murdered man, but the name across the bottom is Danny Stevens - the policeman's real name.