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)
Factual error: Superintendent Kane wears the insignia and uniform only worn by the Commissioner of the Metropolitan Police, not only six ranks above the rank he is supposed to be, but in an entirely different police force.
Factual error: At the end of episode 6, we see Alain crossing the German French border in a European registered right-hand drive Citroen CX. Then at the very beginning of Episode 7, they check into a hotel and the Mercedes parked in the forecourt was also right-hand drive. (00:01:00 - 00:01:30)
True Crime - S1-E1
Other mistake: When Tim Kono enters the lift, he presses floor 8, but gets off at floor 9.
The M.O.D.O.K. That Time Forgot! - S1-E2
Character mistake: M.O.D.O.K. refers to Hoobastank as "the Beatles of the late 90's." Hoobastank only released one album in the 90's, and they didn't become popular until after they released their second album in 2001.
Factual error: The Chief Constable of Dyfed-Powys Police is depicted as a mixed-race woman named Tyler. At the time, the chief constable was actually Terry Grange, a white man. The only woman to ever head the force, temporarily in 2012 (after the period covered by the series), was Jackie Roberts, who is also white. This is a factual series covering real events, not a work of fiction.
Episode #1.1 - S1-E1
Continuity mistake: When Mark got into his cell, the top bunk had a vinyl-like covered mattress (or folded gym mat) that had a lengthwise seam along the middle of the side. He was not fixing his bunk bed while he was talking to cellmate Bernard, but when he was removing items from his bag on top of the bunk bed, it was now covered with a blue blanket and a sheet folded down by the pillow could be seen. A little later, the blanket was also covering the foot of the bed. (00:13:15 - 00:15:03)
Audio problem: After Kate shoots an arrow at the mafia guys, she says "told ya." Then we hear Clint/Hawkeye say "yeah", except he's on camera and his mouth doesn't move, he doesn't even change his facial expression.
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.
Visible crew/equipment: At the end, when Aunt Vi and Delilah return home, just as Miles walks into the house, the green T-mark is visible on the floor.
Planes, Trains, and Automobiles - S2-E1
Continuity mistake: S2/E1 Planes, Trains, and Automobiles: Ashley smoked her cigarette down to the filter with no white cigarette showing but in the next frame there's white cigarette showing. (00:11:23)
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.
Visible crew/equipment: After Raines shares details with Kellett about Semien's crew, when it cuts to Semien waking everyone up, just as he announces his contact with Belgian Federal Police, there's a visible T-mark taped to the floor behind Andre's cot. (00:24:25)