Factual error: A thief has stolen 75 million dollars. He runs out with two large duffel bags. All US bills weigh a single gram. This means that $1,000,000 in $100 bills - the largest bills in circulation - weigh around 10 kilograms, or a little over 22 pounds. $75 million would be 1,650 pounds. No way this could be carried in two duffel bags.
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.
Episode #1.1 - S1-E1
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)
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)
Visible crew/equipment: Camera equipment is visible in the glass of the plane as the shot pulls in on the cockpit. It is fast but noticeable.
Continuity mistake: At breakfast the apple being cut for Camille as she poses a question to Adora is suddenly on her plate as the scene cuts to Adora's immediate reaction. (00:05:07)
Episode #1.6 - S1-E6
Continuity mistake: The paramedics zip up a red body bag containing Helen, who was killed in the fire. But they wheel out a black body bag a few minutes later.
Episode #1.2 - S1-E2
Factual error: The Fitzhuberts' soirée in February 1900 is said to be the first of the new century. The beginning of the 20th century (unlike the Millennium) was celebrated correctly in 1901. 1900 was correctly understood to be the last year of the 19th century.