This is a bit of a mix of Broken Aesop and Family-Unfriendly Aesop, but the moral of Birds of Prey: The Battle Within, the arc from issues 76 to 85, appears to be the fairly stock aesop of "You should accept your friends for who they are and not try to change them," except that what Oracle was trying to change about Huntress is her tendency to kill people. In the end, Oracle apologizes to Huntress, and, in the Dead of Winter story arc (issues 104-108), actually tells Huntress to use deadly force against the Secret Six if she thinks it appropriate, possibly making this the Family-Unfriendly Aesop that sometimes killing people is a good idea.
Marvel's Civil War storyline featured the superheroes favoring registration fighting the superheroes opposing it. Apparently, the two sides were supposed to be presented evenly but due to the clear Aesops of the last century saying that secret identities are good and government oversight of superheroes is evil, it was hard to sympathize with the Pro-Regs. Especially since Iron Man, the Pro-Reg leader, became a borderline Fascist Nazibot for most of the storyline. The whole thing was basically a titanic Idiot Plot where everyone held the Conflict Ball.
The X-Men in particular stayed out of the entire debate surrounding the Super Human Registration Act since in their own comics, government registration of mutants was always portrayed as the first step towards state-sponsored internment/genocide of anyone with an X-gene.
In JLA: Act of God there is an underlying implication that the superheroes were being punished for their arrogance. Even though people like Superman and Wonder Woman are fairly humble in normal continuity (not to mention all the characters who are nothing but humble, such as Captain Marvel), while Batman in this story is ego tripping and denigrating the contributions of his formerly powered friends as they kiss his ass. Apparently the writer thought the superheroes WERE being arrogant because they weren't bowing at Batman's feet and worshiping him as the greatest superhero of them all. This requires the aesop not to apply to Batman, because he's the single most arrogant person in the story.
Also, there's a comment about Supergirl's actions as a superhero was an abuse of power, acting outside the law, only to disregard this and comment about how inefficient police work is and how much more effective she was as a superpowered vigilante and how she should go back to being a vigilante. Then it breaks that aesop because standard police work (like forensics) did more to uncover what happened to The Atom than vigilantism.
Marvel AdventuresSpider-Man # 39 has a foreign exchange student named Kristoff show up at Peter's school, and make a speech about how, unlike many of his countrymen, he doesn't hate America. Peter shows him around, and they talk until it's revealed that Kristoff is from Latveria, home of Doctor Doom. Peter freaks out a bit but accepts him for it. Then the Fantastic Four show up, attacking Kristoff seemingly just because of his Latverian origin, calling him a "potential threat to national security", and taking him away. So, it turns out that he's just a normal, nice kid and the Aesop is that ethnic prejudice is wrong, right? ... well, no, because it turns out that he was Actually a Doombot, and Spidey and the FF have to beat him up. So, the Aesop is that you should never trust people from enemy countries, even when they seem to be perfectly nice, and that it's totally logical to seize and search people who might be a problem.
Mystery in Space #8 featured the story "It's a Woman's World!", in which a man stuggles to prove himself in a future where women have dominated society. It appears to be a story about gender equality, but then you get to the very end, in which the male protagonist forces his wife to do housework, and the wife in turn admits that women ran things long enough and that men should take over again.
One More Day breaks the aesop that Spider-Man is supposed to embody, as instead of taking responsibility for his actions, he dodges it by making a Deal with the Devil against the wishes of its main beneficiary and guilt-tripping his own wife into going along with it. However, in One Moment in Time (popularly known as OMIT), this is retconned so that Mary Jane is the one to have made the deal. Word of God is that the aesop is meant to be "It's heroic to do whatever you can to save a life" but to readers, rewriting history just to save the life of a single person who, in addition to wanting to die anyways and was telling you to let go, let's face it, is likely to die of old age in a few years is simply asinine. The message then becomes "the ends justify the means", and that instead of learning how to cope with loss and move on with your life, you should hold on to what you have and never let go, even if the cost of doing so might be too high; for you and for others.
What makes this even worse is that the "whatever you can to save a life" wasn't selling their souls, give any kind of favor to the demon, or even their love, but he wanted them to give up their marriage. So Peter Parker had to face the consequences of... a chance to get back with Black Cat.
Even worse in that Word of God states that the reason he had the marriage nullified through deal with Mephisto, instead of divorce, is that he felt that having Peter and Mary Jane get divorced would send a bad message to the kids. So... selling your soul to the Devil is a BETTER moral choice than getting a divorce?
The best part of that? Years before One More Day Peter and Mary Jane were ALREADY divorced for some time.
The Sandman has a rare in-universe example. The Kipling-quoting "Indian Gentleman" tells his companions a tale he hopes will "prove" that women are inherently evil in "Hob's Leviathan." But the men in the story are no angels themselves, and as his shipmate Hob points out, the sum Aesop of the story seems to be more along the lines of "men and women are both capable of deeply hurting each other."
Superman at Earth's End has one of these. Superman loses (most of) his powers and has to rely on a gigantic machine gun to solve his problems. After using his gun to kill two Hitlers and a Batman zombie (don't ask) he tells his allies (some little kids with guns) that he's dying. The little kids then bawl and say that guns killed Superman before throwing all of their guns into a bonfire. Of course, no one bothers to point out that guns also saved the kids from two Hitlers and a Batman zombie!
Worse is the fact that Superman himself admitted that the only thing keeping all these kids alive in this After the EndGOTHAM was those guns.
Batman, for God knows what reason, has the gun that killed his parents proudly displayed in his batcave - it's even labeled "THE GUN THAT KILLED MY PARENTS". For someone who despises guns like Batman does, it makes you wonder why he spent time locating the gun in the first place.
Uncanny Avengers is the first real attempt at a team book that has the Avengers forming an alliance with the X-Men in the name of promoting diversity and tolerance for the mutant condition — and in some eyes, it's fallen flat on its ass. Issue #5 has Havok give a speech that could be taken to say, "I want to be seen as more than just 'that mutant'"; however, given the wording, many have taken it as saying, "Merely adopting a cultural identifier such as 'mutant' is a divisive gesture that separates us from others." It's not helped by Issue #9, which features such greatest hits as "Members of the majority don't understand why minority puts so much stock in cultural identity" and "Being born with a certain condition isn't a real cultural identity."
Despite trying to lecture the world about how great mutants were and how they should be allowed to embrace their identities, Xavier spent most of his life masquerading as a normal human who just happened to be a mutant expert. Xavier only involuntarily "outed" himself during Grant Morrison's New X-Men run when he was possessed by his evil twin.
James McAvoy said he actually kept this in mind while portraying Xavier in X-Men: First Class. He pointed out that Xavier is a well-meaning, but ultimately misguided liberal, as he still has tons of societal advantages given that he's white, heterosexual, male, and extremely wealthy. He certainly doesn't have to put up with the same bigotry many mutants face (hell, the same bigotry many real world minorities still face), which causes his message of peace to ring false to many.
Also, compare Xavier's powerset to those of characters like Rogue, Toad, or Cyclops. Xavier has telepathy — a power that he can control perfectly, that has absolutely no negative effects on him physically or mentally, and that is a massive benefit to his life. In comparison? Rogue's powers kill anyone she has physical contact with. She cannot control this or stop it in any way, and has resigned herself to being isolated from her peers. Her powers have drastically injured her self-esteem and social life. Toad's mutation turned him into an ugly, lizardlike humanoid and made him the subject of severe bullying from other children. Cyclops projects a continuous wave of destructive energy from his eyes and relies on special glasses just to live a normal life. Even Phoenix, another telepath, is often overwhelmed by the thoughts of others — to the point of mental instability. Looking at the general trend of mutant powers, it's hard not to think that Xavier really lucked out where the Superpower Lottery was concerned.
Also, after decades of using mutants as a metaphor for an oppressed minority that we should love and respect, Joe Quesada mandates the Decimation event, in which a vast majority of the Marvel universe's mutants are depowered and there are in the low three digits of mutants left. What made it worse is the justification. Quesada claimed Marvel had to decimate the mutant population because Grant Morrison had established that there were millions of mutants across the globe, and that in Quesada's eyes, that meant the Fantastic Racism element no longer worked. Specifically, he pointed out that mutants were supposed to be victims of bigotry, and yet had their own neighborhoods, culture, music, and even languages. In real life, many minority groups have all those things and yet still suffer discrimination from the majority, meaning there's no reason mutants couldn't have numbered in the millions and still have been targeted by normal humans. Quesada's explanation is tantamount to saying that black people, Asian people, Latino people, or LGBT people no longer experience discrimination because "Hey, at least they have their own neighborhoods and pop culture figures!"
In continuities as old as Marvel and DC's, the inevitable retcons often break initially intact aesops. For example, many of the older X-Men storylines involving Nightcrawler made it anviliciously clear that Fantastic Racism is bad, that we shouldn't judge people by their external appearance, and that having a tail doesn't necessarily make you the Antichrist. Enter Chuck Austen, and it turns out Nightcrawler really was half-demon all along.