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)
Factual error: The Saturn in the show is an Ion which stopped production in March of 2007. Therefore, it couldn't be a silver 2009 Saturn they were looking for.
Episode #1.6 - S1-E6
Factual error: When Commander Anne Sampson appears in her body armour she wears the rank insignia only worn by the Commissioner (a crown over a pip over crossed tipstaves in a wreath) instead of that worn by her own rank of Commander (crossed tipstaves in a wreath). She has apparently been promoted four ranks.
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.
The Strangers Across the Street - S1-E2
Factual error: Ed Wiggins and his media liaison are taken by surprise when they see the "Sinful Messiah" article in the Waco Tribune-Herald. The ATF had actually been aware of the article and tried to stop its publication.
Factual error: Luka's military officer cap isn't a Russian cap, it is a Hungarian police cap. The Russian cap is bigger.