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)
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.
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)
Suggested correction: I don't see this is a problem due to the fact that we can't be sure what really happens as apposed to what only happens in Arthur's mind. So if the whole TV show appearance is just another fantasy, he would have skipped the who screening process.
You are free to treat the whole movie as something where things don't make sense because in the fan theory of your liking it's all meant to have subtle hints that the movie is all a fantasy, but the movie does not present that particular talk show scene as a dream sequence. It'd be silly to nitpick the logic in the scene when he is picked from the audience by Murray at the beginning because it's obviously presented as nothing more than his fantasy, but his appearance on the show is what the movie built up to up to that point and is not treated as a parenthesis where logic should be suspended, nor disproven like the scenes with his girlfriend standing in.
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.
Factual error: It is established that Penny Fleck adopted Arthur and that he's been abused. In her file, when Arthur reads it, you can see that she was admitted the first time to the psychiatric hospital at 15 years of age, had multiple episodes with drug abuse, and the file mentions she is 25 and single on the date of the report, 11-2-1952. A single parent already had rather slim chances to adopt in the 50s, but a known mental patient and drug abuser, not a chance. (01:13:40)
Suggested correction: She could have bribed her way into adopting a child. Someone who is desperate for attention could find ways to get what they want.
Suggested correction: It is not firmly established that Penny actually adopted Arthur - in fact, it's strongly hinted at that Thomas Wayne forced her into signing adoption papers in order to cover up Arthur's true parentage.
The established, as in recognized, backed up by documents, 'official' version the main character finds out and acts by, is the one contained in the report, newspaper clippings and flashback; son abused by the boyfriend of an adopted mother. Such story is impossible the way it is presented the moment we see details in a document that overblows it painting this 'adoptive' mother as single and with a history of drug abuse since 15 years old. Penny is not eligible to be an adoptive parent, and yet nobody seemed to have raised an eyebrow about that. If you want to assume that rather than being a mistake with overzealous details in a prop (check out of the original script of the movie, which has none of this ambiguity) whoever arranged the fake adoption documents kinda forgot to also make quietly disappear the mental and medical record invalidating their own fabrication, sure, do that! It's not exactly a small oversight - and really one would wonder why Wayne kept his bastard son with her at all.
Arthur is not Thomas Wayne's son. That was all in Penny's head.
Stupidity: After one of the policemen decides to jump over the railing and right into the angry mob (!), Arthur just easily sneaks by ducking under it and takes a nice stroll that will lead him through an unlocked door. Nobody in the mob he is part of decides to do the same, and you can also see that one of the policemen is turned towards him, but does not even yell at him or move. And of course, with the theater packed with the Gotham elite basically under siege by a mob and guarded by the police, the door is unlocked and unchecked. Why not. (01:02:55)
Suggested correction: The point is they were all too distracted by the tussle to notice Arthur ducking behind the barrier. No cop sees him. The angry mob is controlled by the barrier and not all that large so they haven't taken extra precautions to keep the mob at bay, yet. The door Arthur gets in is probably a fire escape and can't be locked for safety reasons.
I think that with an angry mob worth putting barriers and a big police dispatch, they'd tend to lock the door that is like a 20 feet of walk in a straight line. I mean, they have barriers in front of the stairs, but at the base of the stairs there's an unguarded, unprotected, unlocked door. It's just funny. Not even something in the back or around the corner, no; literally one step to the right of the blockade.
Plot hole: Given Arthur's mother was placed in an asylum it seems unlikely she would've been allowed to keep him once released. The state would've placed him in another home.
Trivia: The social worker who talks with Arthur has the last name Kane as a reference to Bob Kane creator of Batman and the Joker.
Continuity mistake: After Arthur took off the stolen jacket and hat when talking to Wayne he places them in the sink, but in the next cut they disappear. (01:05:00)
Suggested correction: You don't see the sink containing the clothes anymore after he puts them down.
Suggested correction: They purposely stuck the images in a way that it repeats itself over and over again. It's intentionally made in a way of not flowing straightforward.