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.
Awesome Art: Say what you want to about the writing, but the art is amazing, both Lee's interiors and Miller's covers. The art is what's been selling the book. All the critics who absolutely hated the writing said DC could do the fans a better service by reprinting the books with blank speech bubbles.
Crazy Awesome: What a lot of Batman's actions appear to be intended as.
Darkness-Induced Audience Apathy: When both your protagonists are at best sociopathic jerks without sincere empathy for other human beings, it's hard to feel genuine sympathy.
Designated Hero: Batman. He's psychotic, unstable, an all round Jerkass, who kidnaps Dick Grayson, gleefully commits murder, and barely does anything heroic, and yet somehow the reader is supposed to side with him.
Fan Dumb: Ignoring the chronological impossibility of events in the first few issues and claiming that they CAN be explained logically.
There's also the absolute insistence that it absolutely must be a parody and that Frank Miller is just having a laugh at everyone's expectations of him, when there's little to no supporting evidence that the comic is intended as Self-Deprecation, and practically all of his more recent works have turned out as badly, if not worse (see Holy Terror).
Fan Nickname: He is called "The Goddamned Batman" to distinguish him from his mainstream counterpart. In his text reviews, Linkara ends up calling this version BINO, Batman In Name Only. In the video review, he decides that even that name is too Battish, and dubs the character Crazy Steve, a random insane hobo who found a Batsuit somewhere, killed Batman, and then stole Batman's life. Later after seeing Wonder Womanalso acting way out of character, he dubs her Bonkers Betty, an escaped mental patient who found a Woman Woman costume and trench coat, and now runs around hating men. He also gives one to Black Canary.
Harsher in Hindsight: The 1994 Batman/Spawnnote also written by Frank Miller crossover had Batman riled up at Spawn's lethal methods, vexing him to the point where he chucked a batarang in his face. At the time it was understandable; in Batman: The Dark Knight Returns and Batman: Year One he never used lethal force. After his child abusing, cop killing antics in All Star Batman his attitude towards Spawn killing a Complete Monster seems flat out hypocritical.
Hate Dumb: While ASBAR has many faults, some criticisms it gets are simply ridiculous:
"This is nothing like New Earth's Batman!"
Declaring plot points from the first issues that are to be explained later (this is supposed to be a 16 issues series) as plotholes. This said, there are more than a few legitimate examples which can't be explained quite so easily.
The love scene between Batman and Black Canary also count. In the story, they have make out while keeping their mask on. The scene might be Squick, sexy, or even outright funny to some people. Fast forward three years later, Batman and Catwoman have SEX with their mask on. The scene is here if you want to check it out(NSFW)
As Linkara pointed out, Frank Miller once said he considered Batman the most pure and good of all DC superheroes. Fast forward to this comic and suddenly we have the Goddamn Batman smirking gleefully as he slams the Batmobile into cops, abducts Dick Grayson, Age Twelve, and generally runs around acting so brutally out of character Linkara dubs him 'Crazy Steve' because even Batman In Name Only is still too much like Batman.
Memetic Molester: It's often been pointed out that the way Batman treats Robin can get really uncomfortable. Says he's been watching him before Robin was orphaned, takes him away without the kid's consent, smiles as he thinks about putting the kid through hell...
Padding: The series suffers from this. One critic noted the book felt like Miller was spreading 4 issues of story across 20. To put it in perspective, Batman meets Dick Grayson in Issue 1. They arrive at the Batcave in Issue 4. The time in between (the entirety of Issues 2 and 3) is focused on either inner monologue which repeats itself or scenes focusing on other characters (despite this being a book about Batman and Robin). Black Canary's introductory scene takes up half of Issue 3, but all that happens is her getting harassed and her beating up a room full of people.
Strawman Has a Point: When Green Lantern confronts Batman, he points out that Batman is hospitalizing people with his excessive violence and endangering a boy as young as Dick.
They Just Didn't Care: Frank Miller has become so obsessed with making Batman "better" than the police or any other superhero, that he's stopped caring about making Batman actually do anything heroic.
During the story, Dick Grayson's face is on a Milk Carton... While he's still being driven to the Batcave, meaning that either the trip takes several days, or Batman's car drove into a time warp.
The Batcave has a wall of weapons that Batman in any normal story would never use, including guns in it. That's all.
Other superheroes such as Black Canary, Plastic Man, and Wonder Woman appear in the comic, and Frank Miller writes them all as psychotic murderers with no regard for who their mainstream counterparts are.