Factual error: This is set in 1933. When the killer comes back to his lodgings in episode 1, Lily on the stairs is humming "Somewhere over the rainbow", which was written for The Wizard of Oz in 1939. She also happens to wear bright red shoes, which feels like yet another Wizard of Oz reference. Interesting enough, the subtitles say that she is humming a different song, "Night and day" which she openly sings throughout the series and was already a hit in 1932. (00:44:10)
Plot hole: The FBI is looking for a "very tall" guy who has been identified as such by various witnesses and by their own expert due to the entry angle of a fatal bullet wound - 6'5" at least. They send Elena into a bar near the end of the show to meet the Albanian perp who when he arrives is barely average height.
Character mistake: When the bodyguards enter elevators, they always enter last; they should be entering first to ensure it is secure. This is particularly visible in the hotel where the elevator can stop on any floor and Special Protection Branch does not have complete control of it as an express.
Visible crew/equipment: During the strobe lighting at the very end of series three, with Faustus Blackwood seemingly releasing something evil, it is noticed that during the flashes, it is leaving after effects, showing the shape of the equipment that make the strobe lighting. (00:58:00 - 01:00:00)
Revealing mistake: When Harrow and Nicholls are in what is supposed to be Queensland Police Headquarters, a window is visible with logos for Radio National and Triple J behind them. This shows that the scene was recorded at the offices of the Australian Broadcasting Corporation. (00:31:04)
Other mistake: The shopkeeper hands Danny a Beretta 92fs/M9 handgun with a detached suppressor. To mount a suppressor to the muzzle end of the handgun the barrel needs to be threaded, and this one isn't. Sometimes barrels are threaded and when no suppressor is attached they are protected by a thread barrel cap to protect the threads from dings and bumps. There is no thread cap here either. (00:22:35)
Continuity mistake: As the US Embassy in Caracas, Venezuela is being evacuated, the ambassador steps into a Black Hawk helicopter with markings 33 and 3016. After her Black Hawk arrives at the Aeropuerto Francisco de Miranda Airport in Caracas, it has markings 60 and 2065. (00:25:48 - 00:29:20)
Factual error: Sir John Franklin, Francis Crozier and James Fitzjames all wear a crown over an anchor on their epaulettes, the rank badge for a captain with over three years in rank. This is correct for Franklin and Crozier, but Fitzjames only held the rank of commander, and should therefore only be wearing an anchor on his epaulettes.