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.
Continuity mistake: In the scene where Debbie clears everything off the mantle, she leaves the bronze baby shoes and turns to talk to Marty. Then we see over her shoulder that the shoes are gone. (00:04:30)
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.
Character mistake: At Houdini's performance for the Kaiser, a German officer with the rank of Lieutenant Colonel (Oberstleutnant) is asked to assist. Houdini at first correctly addresses him as Lieutenant Colonel, but then incorrectly refers to him as Lieutenant several times after that.
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 One - S1-E1
Factual error: In the fairground chase the characters jump over some Heras fencing that wasn't invented until the next decade.
Plot hole: After Jack's wife disappears for a couple of days, he receives a call from her cell phone from the cab driver who found it. Jack arranges to pick it up from a hotel. After he picks up the phone, its battery dies as he checks it, at 12:13pm. In the next scene with Jack, he's in a bar with the phone plugged into a charger; when it comes on, he checks it - the battery shows 3%, but the time is 9:13pm. He apparently wasted 9 hours when he's in pursuit of his wife's mysterious disappearance before plugging in the phone. A scene later, the time is now just a minute later at 9:14pm but the battery is now charged to 79%. (00:37:25 - 00:40:45)
Broad City - S5-E10
Continuity mistake: When Ilana goes to flag down a New York City taxicab on the street, there is a Honda Civic parked behind her with a yellow New York license plate. After the camera pans the opposite direction to show an approaching taxicab, it pans back to Ilana and the same Honda Civic - now sporting a different white New York license plate. (04:40:00)