Factual error: The show starts in 1901 with a woman constable working for the Metropolitan Police Service (Scotland Yard). However women didn't join the Metropolitan Police until 1919 (which is also when the first woman's police uniform was created). Prior to that, women volunteered in the Women's Police Service, but that wasn't formed until 1914.
Factual error: With two exceptions, all the policemen shown in both 1949 and 1953 are wearing the old-fashioned tunics with high, closed collars. The Metropolitan Police adopted the more modern open-necked tunic with collar and tie in 1948, and this is shown worn by the policeman who visits Christie and by the officer who arrests him.
Factual error: When Riggs and Murtaugh go to investigate the scene where the body of Ramon Alvarez is discovered Scorsese states that Alvarez's dog tags say "SEAL" on them. His dog tags would not say that he is a SEAL on them. The information listed on dog tags is as follows: Name, Social Security Number, Blood Type, Service Branch, and Religious Preference. There is no way to tell what a given service member's MOS (job) from his or her dog tags. (00:14:00)
Factual error: Cars in Thailand drive on the left side of the road with the steering wheel on the right, unlike the ones shown in the episode.
Factual error: You can see the Unicredit Tower on the Milan skyline - the episode should be set in 1997 but the unicredit tower was built in 2012. (00:31:00)
Factual error: Almost all the people being shown skydiving have no experience doing it, but the instructor allows them to jump out of the plane unassisted after only giving a theoretic introduction into the subject. First time skydivers are always assisted in some fashion, usually through a tandem-jump or a parachute opened by a line attached to the plane.
Chapter Seventy-Nine: Graduation - S5-E3
Factual error: Archie revealed that he joined the Army to avoid repeating his senior year and was shipping to Basic Combat Training the following morning. Before leaving he would need to wait for his background and medical checks to be completed (including getting his dropped murder charge reviewed for suitability), which could take weeks. Also, the Army usually requires a diploma or GED to ship to basic. There were brief periods when the Army was accepting dropouts but it was not when this episode took place.
Factual error: When they deliver the drug and guns to the reservation, the car James is driving, has an Arizona license plate on the front of the car. Arizona does not issue plates for the front, only one plate for the back of the car.