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)
Continuity mistake: During the opening credits, while the main character is clowning around with the sign the same passersby (in particular two women - one with a blue coat, the other with a plain light brown one) keep walking past him in multiple shots. (00:01:35)
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)
Continuity mistake: Just before Arthur is hit with the yellow sign there is a person behind him that disappears between shots. (00:02:45)
Continuity mistake: When Arthur Fleck gets hit by the yellow sign, the sign breaks in two different ways in the two separate shots - the first time it is evident that the bottom half of the sign splits further in the middle, while in the wider angle it stays largely intact. (00:02:50)
Continuity mistake: When he gets hit by the sign, the round spinner on the back of the sign disappears. (00:02:50)
Revealing mistake: The sign with "everything must go" breaks a second before it hits Arthur. (00:02:55)
Suggested correction: The sign with "everything must go" breaks at the exact same second as it hits Arthur, and there is no point to make it somehow break by some weird special effect without any sense.
Continuity mistake: When Joker is beaten up at the beginning of the movie, he changes position on the ground between shots. The pieces of the sign next to him also change position; in the last shot that lingers till the movie title pops up, the part with the letters "ST GO!" is face up, it was not in the previous one. (00:03:05)
Continuity mistake: Talking with his social worker Debra Kane, Arthur is moving his legs nervously as she asks him about the journal. In that shot his cigarette is almost smoked to the filter, but in the next one there's still an inch to go. (00:05:35)
Continuity mistake: During the scene when Arthur is making funny faces at the kid, the hair wisps on Joaquin Phoenix's forehead change position between shots. The mother also turns slightly towards him in almost every cut, but his antics are in continuity. She keeps 'just noticing' him and turning towards him without the opposite movement to balance. (00:07:45)
Factual error: Heading home after the first encounter with the social worker and the incident with the kid and the irritable mom, Arthur is trudging to get his meds at Helms' Pharmacy. Parked in front of the pharmacy there's a sedan, a Mercury Grand Marquis, which by bumper design is a post 1988 model, way past the events of the movie, in 1981. (00:09:20)
Continuity mistake: When Arthur imagines that he is on Franklin's TV show the button on Murray's jacket is off then on. (00:12:20)
Continuity mistake: The first time we see Randall, he hangs his clown suit next to his locker; his right hand drops, but in the next shot it starts still up. (00:15:50)
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)
Continuity mistake: In the close-up of the gun at the beginning of the scene when Arthur is watching "Slap that bass" from "Shall we dance", notice in background the bullets; there is one angled parallel to the gun and 3 more at a slightly different inclination, plus one at right angle with the first (plus other 2 not quite aligned with the rest). A couple seconds later he aims the gun at Fred Astaire, and that group of bullets changed angle entirely. (00:22:10)
Continuity mistake: Arthur inadvertently fires a gunshot in his living room. Right as he tumbles across the table, he knocks off the table the paper bag the gun came with, and knocks towards the edge of the table a pill bottle. In the closer angle shot when he cranks up the volume of the TV set, the bag is still on the table and the bottle is gone. The newspapers are also positioned differently. In the next shot we're back to the previous state (pills up, bag down, newspapers closer together). (00:23:10)
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)
Suggested correction: By the directors own admission, the date the movie is set in is never mentioned, nor is there any mention of a real city it is set in. This movie is set in Gotham City, a city that exists only in the Joker universe, where this paving could have been invented years earlier than the corresponding year in our (real) universe. This is more of a trivia than a mistake.
By the director's own movie, everything about the setting is specific to the early 80s. It's a marginal part of the urban scenery that they didn't find important (or did not think of, it's not exactly obvious) to fix for consistency. I don't see why we have to think that a movie that deliberately puts real life advertising, technology, aesthetics specific to the 1980s (Philips even mentioned specifically in interviews that he had in mind New York City of the year 1981) and flaunts the marginalization and cruelty of society would encourage leaving in deliberately something that improves quality of life for the handicapped. It's the classic mistake of something not supposed to be there that needed to be covered but was not.
Continuity mistake: Arthur is taking notes during the standup comedian's performance. There are a cigarette lighter and a packet next to him on the table; in the close-up that follows, they have changed position. (00:25:45)
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)
Continuity mistake: Right at the beginning of the dance in the children ward, the kid with a blue hat on Arthur's right is sitting cross-legged with the hospital robe covering his calves and feet. At the first cut, his jammies and socks become visible. (00:28:05)
