You Always Hurt the One You Love - S1-E1
Factual error: In the opening sequence, Dr Frank Winter is hitting golf balls in the desert while a radio announcer is reading off names of Americans killed in action. One name is identified by the rank of "Lance Corporal," which the US Marine Corps first adopted in 1958, long after the war. Several minutes later, Dr Winter dresses down an Army MP, referring to him as "Sergeant First Class." During WW2, that rank was actually "Technical Sergeant."
Episode #1.5 - S1-E5
Factual error: Catherine is told she's been recommended for the Queen's Police Medal for Gallantry. This has not been awarded since 1977, when it was replaced by the Queen's Gallantry Medal, which is still awarded.
Things Were Different Then - S4-E16
Factual error: The car is referred to as a '73 Cadillac Eldorado. It has square headlights, and therefore is a '75 or '76.
Factual error: Helmuth von Moltke is addressed and referred to several times as Field Marshal. He wasn't promoted to that rank until 1871. He was a lieutenant-general during the Second Schleswig War in 1864.
Factual error: Constantine's driving licence is shown - it's a UK licence, but no. 7 where his signature should be is blank, and no. 8 where his full address should be simply says "England, United Kingdom." Both are printed onto the licence when it's issued - without that info it would never be issued in the first place. And it's not a fake, because why would he have a fake licence with his real information on it? It's just an incomplete prop.
Factual error: Congressman Tate and his brother are on the shooting range. They are wearing high-tech electronic earmuffs that block harmful noise like gunfire while amplifying speech and other low-volume sounds. Although, they are wearing high-tech earmuffs, they both have an ear exposed so they can hear each other talk. (00:13:00)
Sea Change - S3-E1
Factual error: President Dalton's memorial US Flag in his office is folded wrong and displayed upside down in the flag triangle. The US Flag is always displayed with the stars pointing up. (00:47:00)
The Alchemy Murder Case File 3 - S1-E8
Factual error: The Latin text on the supposedly old tome (which is written with a modern font) displays an alchemist diagram. It is a reproduction of a famous drawing meant to portray the Philosopher Stone in the Sylva Philosophorum, but whoever copied it does not know Latin, because certain words are misspelled ("Merceralia" instead of "Mercurialia", "Humides" instead of "Humidus"). (00:10:10)
Factual error: Eve wears an imaginary cap badge. It certainly isn't Royal Military Police.