These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Alternative Character Interpretation: Were Chief Rojas's reasons for going after Batman legit because he believed Batman to be a menace or was it because of something else? Depending on your views, Rojas might've been a Dirty Cop. He did once say that about Ethan Bennett that "He never knew how to play ball." Plus there's his timing of partnering Ellen Yin and Bennett up shortly after Thorne was defeated and some of his tactics in hunting Batman were similar to the comics version of Gillian Loeb.
And Strawman Has a Point. Despite his idiocy and stalwart hatred of Batman beyond all reason, Rojas is correct that Batman is a vigilante and should be persecuted like the rest of the Rogue's Gallery in episode 12. His reasons for wanting to do so aren't so golden, but vigilantes are criminals. It doesn't help that in the very next scene that episode, Batman's fight with Joker's thugs causes about as much property damages as Joker himself does.
Base Breaker: The Joker's dreadlocked design constantly gets heat from Batman fans outside the show's fandom, having been nicknamed "Rasta-Joker". Among people who enjoyed the show, though, he's widely praised.
Complete Monster: The Joker was once a man who wanted to make people laugh, but after he fell into a vat of chemicals, he became an insane, murderous Monster Clown and Batman's Arch-Enemy. Devoid of any empathy, Joker commits crimes he views as "jokes" at the expense of Gotham. Terrible things he's done include poisoning people with his deadly laughing gas; putting people in various death traps; torturing Detective Ethan Bennett for hours and causing Bennett's mutation into Clayface (for no reason other than to see what it would do to him); impersonating Batman and gassing people for minor crimes; using Bane's venom to go on a rampage; attempting to drop a teenage boy into a vat of chemicals; frequently mistreating his henchman and his girlfriend Harley Quinn, as well as abandoning them to be arrested or even to die; and filling the abandoned tunnels and mine shafts beneath Gotham with miles of dynamite to collapse the city to oblivion. That he manages to qualify despite being toned down to all get-out is a testament to how twisted a character Joker is.
Quite a few of the villains got their own fanbases: Hugo Strange for being a Magnificent Bastard. The Riddler for his Dark and Troubled Past, his new gothic look, and being cool as well as voiced by Robert Englund. Clayface got this status for being an outright tragic character as well as his alter ego, Ethan Bennett.
Robin. His introduction was actually one of the reasons so many fans consider the 4th season the strongest.
Fridge Brilliance: Clayface is almost always frowning or appears to be frowning, while Joker is always smiling, bearing resemblance to the "Tragedy and Comedy" masks, symbolized even more by Ethan's life being outright ruined by becoming Clayface, a tragedy if you will.
Fridge Horror: In "Strange Minds", Batman goes into the Joker's head and encounters the pre-chemical bath Joker, "the only shadow of his former self left" in his mind. ...Who then gets dunked into a vat of chemicals and becomes another Joker.
Growing the Beard: Depending on who you ask, this is seen to happen around the beginning of either season 2, 3 or 4.
Moral Event Horizon: Joker's torture of thugs is certainly crossing this, plus what he did to Ethan. And Tony Zucco crosses it in the first ten minutes of his appearance by killing Dick Grayson's parents (before he became Robin) just because Mr. Grayson called the cops when Zucco got a bit threatening.
Replacement Scrappy: Basil Karlo. It really doesn't help that his introduction episode is the last time Ethan even appears.
Strawman Has a Point: The episode "Rumors" involves a new vigilante capturing all of Batman's rogues with the intention of eventually executing them. Batman is naturally against this. While this is a common story for Batman, and his moral code is much easier to justify in this series since the villains have a much lower body count and haven't been at this for years, Batman doesn't really offer any rebuttal beyond "my way is right". It's also pointed out that Gotham didn't have criminals like them before Batman, something he never acknowledges in the episode. Ultimately, the story doesn't do much to disprove Rumors' arguments, and it comes off as "the hero is automatically right".
Tear Jerker: The endings to "White Heat" and "Riddler's Revenge", as well as every episode featuring Clayface.
The Riddler's backstory is especially notable (As well as overlooked) the fact that this Riddler has the most tragic origin of all the animated Riddlers.
Rino Romaro got this as the new VA for Bruce Wayne who wasn't Kevin Conroy. Fans did warm up to him in due course, though.
This is especially notable when you realize how obviously the writers went out of their way to stand on their own, away from the influences of Batman: The Animated Series. By not doing Mr. Freeze's famous origin, making the Penguin his rude disgusting persona instead of his gentleman persona, Joker's design, and Harley Quinn having a completely new origin note but of course still written by Paul Dini
The Untwist: It is virtually unthinkable to not give Mr. Freeze his tragic backstory in any modern incarnation... but this series decided to give a more traditional supervillian origin. Although barring that, this version of Freeze has his own merits as he still isn't outright evil and his powers were appropriately unique compared to other criminals Batman faced.