Continuity mistake: When Joker is dancing on the stairs he throws his cigarette. Then when the cops arrive at the top of the stairs we see him blowing smoke from the same cigarette. Then the cigarette is gone again. (01:30:40)
Factual error: In the beginning of the movie, while dancing with the sign, the guitars in the window of the music store are D'Angelico guitar designs that weren't released until the 2010s. (00:01:35)
Factual error: The cart behind Arthur during his dance for the kids has drawers of different colors. That is color coding reflecting the Broselow Tape, a tool used to measure pediatric patients and give estimates of the appropriate scaling for treatments. It's a system that started only in 1985, while the movie takes place in 1981. (00:28:05)
Factual error: In the elevator, Arthur meets the single mom living on the same floor, Sophie Dumond. In Sophie's shopping bag, the one recognizable food item is a box of Betty Crocker's Hamburger Helper - Potato Stroganoff. Thing is, the packaging uses (look at the design of the 'menu idea') a design predating the introduction of the Helping Hand mascot, which happened in the late 70s. She just bought it, it can't be ancient, long expired food. (00:19:35)
Factual error: When Joker is following his neighbor to her work, the 'no standing' sign is not from the time period. Also when Joker is fleeing from two police officers into the subway at 18 avenue, the subway sign with the red EXIT background is not of the time period.
Plot hole: Arthur's appearance on the talk show makes hardly any sense. The show is a close port of Johnny Carson's Tonight show, for a huge audience, and yet he receives no screening at all, they put him (someone NOBODY in the staff knows the first thing about) on the air literally without a clue of what he is gonna do or say, and wearing a highly controversial costume. And, when he murders Murray, it is implied that everyone was able to see him doing that right away and he is cut 'off the air' at some point, as if the show were really live, which is preposterous for this sort of program outside of specific events (similar to how in contemporary TV "Jimmy Kimmel Live!", is not live). Even earlier when Arthur opened the letter his mom addressed to Wayne, you could hear the end credits of "Live with Murray Franklin" with the announcer saying the show is "Taped live in front of a studio audience." (00:48:00)
Other mistake: In a city of millions and millions of inhabitants, when the protagonist chases his neighbour till she enters the building in William Street, the taxi that passes in front of him as he crosses the road is the same taxi with license plate T 9362 that nearly ran him over at the very beginning when he was trying to get his sign back. Which incidentally is a model of Chevy Caprice post-1981 (likely a 1987 Caprice). (00:02:25 - 00:24:35)
Audio problem: The police officers chase Arthur to the subway. When the subway train starts moving after they just barely got in, the sound resembles the acceleration sound of a R143 subway car, which actually didn't exist until 2001, but it's a much older car, the R32.
Continuity mistake: Murray's blue jacket, when he comes out from behind the curtain his jacket is open, than he makes a twirl and the first button of the jacket is buttoned up.
Factual error: When Joaquin Phoenix and Zazie Beetz are taking a walk after Arthur's performance, between the arcade and the newsstand there's a modern video intercom with keypad, not quite fitting the 1981 setting, since the first model of its kind was introduced in 1984 (kinda odd to leave something like this in when they went through the trouble of placing appropriate arcade posters really close by). (00:45:25)
Factual error: Following Zazie Beetz, Arthur arrives in front of the bank. The crossing is using red colored tactile paving. While technically already invented, truncated domes paving was not adopted in the US in the early 80s, but began appearing in the early 1990s at public transportation stations, and it was not until 2001 that they were used in curb cuts. (00:24:35)
Factual error: The ambulance that crashes into the police car Joker is riding in towards the end of movie has the styling of a contemporary vehicle, not one from the early 1980s.