Lampshaded in the first JLA Classified story, which revealed the Batcave has a "sci-fi closet" full of rayguns, teleporters and antigravity discs. He just hates using it.
By comparison, Batman has used Powered Armor in stories that take place in the future (Batman: The Dark Knight Returns, Batman Beyond) where his armor is a contemporary innovation. The Batman Beyond armor was over 20 years old and thus dated (but still effective) by the time Terry McGinnis took up the role.
Gotham's status as a hellhole, Arkham Asylum as a Cardboard Prison and the existence of Joker Immunity have all been lampshaded / Hand Waved as due in part to a supernatural curse, which Batman's magical allies (Zatanna, The Spectre) do not seem to know about, let alone consider removing. In one case it was even Etrigan sealing demons under Arkham and he never told Batman about it.
Likewise, it is frequently established as the most nightmarish city on Earth, a metropolitan hellhole with a ridiculously high violent crime rate overrun by murderous psychopaths like The Joker who terrify even the superpowered rogues of other cities, yet Supes and the rest of the demigodic heroes who could easily clean up the place and overpower pretty much all of those homicidal lunatics in seconds barely do any meaningful crime fighting when they visit. They actually show up quite frequently—they don't really "stay out of Gotham"; it's just that on those occasions they tend to talk with Batman more than they actually help him out.
In The Man of Steel, Superman aids Batman with a criminal named Magpie. At the end, Batman says that Gotham requires a different touch than Superman's Metropolis.
It doesn't necessarily work both ways, though: during The Death of Superman series, Batman patrolled Metropolis in the days during and after Superman's funeral. He even played by Superman's rules.
During the No Man's LandBat Family Crossover, the government features a literal version when it cuts Gotham off from the rest of the USA and enacts legislation to prevent anyone (normal or superhero) from going in or out even to assist. This proves remarkably effective given how many superheroes are aliens, or for whatever reason shouldn't care at all about the ruling:
Superman travels to Gotham to deliver supplies and help set up a power plant to provide heat during the winter. The plan falls apart leading to Batman explaining how the city has changed and Superman realizes he's "not up to the task" of fixing Gotham and suddenly leaves without Batman even considering that maybe, just MAYBE, having someone with Superman's level of power would help him bring order to Gotham even faster. Similarly, the Huntress in a JLA storyline points out the League's refusal to assist Gotham and Superman replies her presence is the League's presence. Superman does return to No Man's Land... but only as Clark Kent. He still seeks to help, but as a normal man doing things like growing gardens for food. He even dirties up his appearance to make it look like he's been there all along. While Batman points out that his disguise is flawed (no one in Gotham has smelled like soap in months), he admits it's fine for the people Clark intends to be among. This is more of a desperate Deus Exit Machina from the writers than this trope, but it still applies. (Bruce mentions that there's been a drought on top of everything else, and shortly after Clark leaves it starts to rain, so apparently he can do super-stuff, just not blatantly.)
In the League's own book, it was shown that, during "No Man's Land", they were keeping a slew of opportunists (Kobra's organization, evil Atlanteans, assorted alien armadas, etc.) from seizing Gotham for their own. This neatly balanced Superman Stays Out Of Gotham with "Brainiac Is Kept Out Of Gotham." Of course, they missed a few villains that still managed to get in during the chaos, so they weren’t that good at it.
Jim Gordon revealed during the crossover that he cannot get a job in any police department outside of Gotham as no one wants a cop who needs an "urban legend" to do his policing for him, which Batman is considered despite being a known member of the JLA and living in a universe with other cities that have local superheroes (he mentions Keystone City by name). Of course, it’s extremely hypocritical for the other departments to look down on Gordon for his reliance on Batman, considering how many of these other police departments rely on superheroes to keep them safe and help them catch supervillains. Keystone City is especially guilty in that regard given they have been shown, in multiple storylines, to be nearly useless at dealing with the Rogues without the Flash around. Of course, they are the Keystone Kops.
Barbara Gordon, who was Batgirl until The Joker put a bullet in her spine, was turned into an information-broker for the superheroic community. Yet did not regain the use of her legs despite the loads of superpowers, magic, and technology the DCU had to offer. In-universe, Barbara has justified refusing offers to insta-heal her spine as not wanting special treatment for being a superhero that a regular citizen wouldn't have access to. As of Batgirl (2011), her back is fixed and she's back to being Batgirl, although she feels incredibly guilty about being given a cure that the average person doesn't have access to.
In Elseworld's Finest: Supergirl & Batgirl, Batgirl reinforces this: she banishes all parahumans from Gotham after closing the city off. However, her ban is defied by Supergirl, who flies to Gotham to help whether Batgirl likes or not.
This whole situation was lampshaded in an out of continuity story Gail Simone did for Sensation Comics Featuring Wonder Woman. After Batman and his partners are temporarily sidelined, Wonder Woman comes to Gotham and not only makes a significant dent in its supervillain underworld, but even causes some of Batman's foes to reform in the process.
Superman and Batman have both mentioned having a noted dislike for operating in each other's respective cities. Superman hates working in Gotham because it's so dark and dingy for the sun-empowered hero, not to mention most of the buildings are lined with lead, blocking his X-ray vision, making him less effective than he could be. Batman hates working in Metropolis because everything is so brightly lit, making it difficult to hide and the buildings are spaced further apart preventing him from scaling them easily.
During one very interesting conversation in Justice, Batman observed with some admiration that the reason Superman publicly talks about having X-Ray Vision and other Super-Senses is specifically because it serves notice on the criminals of Metropolis that the shadows cannot hide them from him. This contrasts the criminals of Gotham, where Batman has had to become a part of the night in order to chase down the criminals that use it as cover for their crimes.
In the Novelization of Kingdom Come, Wonder Woman addresses this as well, noting that Superman could have been a Mysterious Protector, but instead flew around the city "like Apollo in the sky," partly to inspire, but also let everyone know he was on watch.
Superman/Batman: Discussed in one issue where it's explained that whenever Batman has business in Metropolis, Superman insists on keeping watch on Gotham City. The Dark Knight always makes sure to return the favor for the Last Son of Krypton. Anyhow, Superman notes that Commissioner Gordon always seems happy to see him. Furthermore, the two have noted in the same issue that crime and trouble come in very different shapes in their two respective cities; with Gotham City, it's mostly about psychopaths like Joker, Zsasz, Firefly, Two-Face, and the rest robbing banks or blowing stuff up. Metropolis, on the other hand, is mainly endangered by the "sci-fi monsters rampaging down 2nd Avenue," or something to that effect.
Another Superman/Batman issue attempts to justify it: when the two characters swap cities for a bit. Batman has trouble operating in Metropolis, where the bright lights and widely spaced out buildings make it difficult for him to do his thing, meanwhile Superman finds that, due to all the lead in the construction and the echoes between the tightly-spaced but porous buildings, his supersenses are essentially useless.
In The Supergirl from Krypton (2004), Batman explains he is out of his comfort zone when he leaves his city and its familiar dangers to go to alien worlds and face evil cosmic horrors.
Batman: I don't belong here. My place is in Gotham. Where my parents were murdered in the streets.
In a somewhat funny if grim aversion, during Battle For The Cowl, where Batman is believed to be dead and the villains of Gotham are all rampaging, Nightwing runs around trying to deal with the problems and quickly decides 'Nope' and calls in 20-25 other superhero allies of his to help him bring order back to Gotham. He probably remembers the last time Batman tried to deal with that situation himself.
Batman (Tom King) seems to be a Deconstruction, as Batman deals with two new flying heroes who state they'll be protecting Gotham, after saving a crashing plane and Bats' life. Bats' reaction? Incorporate them slowly into his fold, seeing what they can do and study their background. He learns they're normal people who have a similar background to him, enough that he's pleased that advice he gave one as a boy has been taken to heart. The same storyline also averts it: When one of the new heroes ends up going berserk due to suddenly coming across the Psycho Pirate, Batman calls in the Justice League to help take him down.
In the DC Universe Holiday Special 2008: A Day Without Sirens, a "Day Without Sirens" is proposed right before Christmas. Commissioner Gordon believes such an initiative is doomed to failure. The criminals of Gotham would never heed such a calling. However, the day proceeds without police sirens. It turns out that◊ Barbara Gordon teamed up with Supergirl and both girls handled covertly all emergency calls during that day. Supergirl ended up completely exhausted, though, making clear she cannot keep it up forever.
Babs: Just rest easy knowing you did something special today. Supergirl: You really think so? Do you think this one day is going to make a difference? Babs: I know so. Never discount the healing power of a little hope, Kara.
There is another, more subtle reason that is danced around as well. Superman and other heroes understand that Batman needs to patrol Gotham for his own mental health. Batman needs to be a protector, and he can't be one if he's constantly being aided.
A Death in the Family has an epic lampshading of this — when Superman foils Joker's attempt to assassinate the UN General Assembly through poison gas, Joker completely freaks out, and calls it "unfair" that Superman is interfering.
Batman/Superman: World's Finest: Defied in the first issue. Poison Ivy decides to make trouble in Metropolis and is shocked when she discovers Batman and Robin have followed her from Gotham City. Later, a flashback scene shows Superman going to Gotham and disguising himself as Batman in order to serve as a distraction as the real Batman saves Robin from the Penguin's death trap.
Conversely, Batman stays out of Blüdhaven, which is his former protégéNightwing's patrol. Dick Grayson has a very different touch than Bruce's, and regardless, Bruce trusts Dick implicitly and doesn't presume to undermine or interfere with his life there.
, Gotham is directly across the Delaware River from Metropolis!)
