In An Alternate Keitaro Urashima, the Hinata Inn serves as this, courtesy of Granny Hina 'helping' its residents by letting them stay there and doing absolutely nothing to try and help with their various issues. As a result, Keitaro wants absolutely nothing to do with the place.
Our protagonist Twilight is a very obviously delusional and paranoid mare, though how delusional she is and whether she has a reason to be paranoid has not yet been made clear...
Rainbow Dash and Fluttershy have their aggressive and shy personalities respectably, turned Up to Eleven with Rainbow violently lashing out anyone she can possibly perceive as a threat to her or Fluttershy and Fluttershy having never spoken in years and on at least one occasion attempted suicide.
Though her personality is mostly still intact, Pinkie Pie suffers deep emotional andphysical scars from accidentally burning her family's house down and almost getting herself killed, trying to bake a cake.
Applejack and Rarity at first glance seem to be just fine, being staff members at the asylum instead of patients and are seemingly happy with their respective jobs. Until we learn that Applejack, being an Earth Pony, is a victim of Fantastic Racism due to having chosen to go into medicine, instead of just staying on the farm and has lingering guilt over having to leave her siblings back home, including Big Mac who due to their parents death, was never given the same opportunity she was. Meanwhile, Rarity's efforts to try to help ponies are put under a harsh light after it is revealed that her little sister Sweetie Belle was lobotomized.
In My Huntsman Academia, three-fourths of the main cast have some kind of Dark and Troubled Past. For instance, Izuku was the target of endless amounts of bullying and scorn, Ruby and Yang lost their mothers, Ren and Nora lost everything and lived on the streets for years with only each other for company, and Shouto was given Training from Hell to fulfill his father's ambitions. This generates heat within each team and with each other as their deficiencies and neuroses clash.
The Fellowship of the Ring has become this in Bag Enders. Understandably, as they've been stuck with each other for six thousand years. Gandalf has become a Dirty Old Man who drinks copiously and spends all his time in front of the TV. Merry and Pippin work a string of low-paying jobs in between drinking as much as Gandalf and shagging Anything That Moves. Frodo is suffering from the incurable Post-Ringbearer Syndrome and spends half his time in a mental hospital or latching onto the long-suffering Sam. Gimli works the night shift and is rarely seen. Aragorn suffered a messy divorce from Arwen and is now the nearest thing the Fellowship have to an authority figure. Legolas is the usual viewpoint character and Only Sane Man, but is slowly cracking under the strain of having to spend eternity with the Fellowship instead of in Valinor, and on one occasion was possessed by the understandably-irritable spirit of Boromir.
Brave New World, a Pokémon fanfic and sequel to Latias' Journey, has a cast chockablock with this trope. Ash Ketchum has a human soul inserted into the body of a Lucario and he has fleeting memories from his life as a human, haunted by memories of previously destroying the world after watching his loved ones get killed; Pikachu is a samurai who knows he may one die in battle; the baby Larvitar, Tiny, had a traumatising birth and refuses to open up to anybody; Lily was born with the purpose to be a sacrifice to Giratina and has been abused all her life by her mother; Dawn was betrayed by her best friend, cannot speak because her vocal cords were removed at birth, and her entire ninja clan was exterminated; and Briney lost his wife to the villains, wearing her eye in place of his missing one. The only major heroes who escape the trope are Leo and Sasha, although they have their own troubles.
The fanfic's version of Misty maintains her phobia of bugs, but it was caused after she was brainwashed into becoming part of a Hive Mind and believing she was an insect, until she was saved by Ash and Pikachu, suffering from a traumatic recovery.
Light: Do you know what it's like to be left alone here with these people? They're all stupid and crazy.
Child of the Storm has Tony Stark all but invoke this trope when Sif's werewolf issues are brought up. Harry and his friends are a particularly good example:
Harry's first memory is of his mother being murdered, and was abused for a decade or so by the Dursleys, with much of his character development being coming to terms with family that actually cares for him and that he's now able to express his emotions. And you know, there's a near death experience or two, attempted murder, that sort of thing. After his run-in with the Red Room in the sequel, it all gets exponentially worse.
Carol has a verbally abusive father with a Stay in the Kitchen attitude, while her mother is implied to be an Extreme Doormat, while also having to deal with a realistic variation of So Beautiful It's a Curse - lots of grown men hitting on her and touching her without asking, and as a result of this and her unsurprisingly abrasive personality, she has few close friends aside from Jean-Paul and Lex Luthor, both of whom have issues of their own (she's actually the latter's Morality Chain, for example, and ends up functioning as one to Harry too, in the sequel).
Her mother turns out to have considerable Hidden Depths, and reasons for her attitude (a difficult relationship with her mother, for a start, who Carol heavily takes after/models herself on), and she and Carol reconcile. Her father, though, is a psychologically abusive dick, who honestly thinks he's doing the right thing by trying to make Carol a proper lady and her quiet, arty younger brother into a manly man - to the point where he's willing to resort to Mind Rape to pull it off, and doesn't even understand it's wrong.
Jean-Paul is Camp Gay and suffers a fair bit of homophobia because of it. Oh, and his sister is basically insane and he's the only person she really responds to.
Uhtred is the youngest of a large and fairly successful family, and consequently has something of an Inferiority Superiority Complex as he tries to be the best simply to stand out.
And Diana is the daughter of Hercules and Hippolyta and thus a target for Hera's wrath. It is further implied that the reason she's living in Asgard - and consequently hasn't seen her parents for years - because Hera repeatedly tried to murder her simply for existing. She's also an Empath and has been since she was a small child, making her involuntarily wise for her years, to the point where Harry specifically notes that her eyes are far too old and knowing for her body.
The most healthy relationship given significant focus in Dominoes so far is the still-distant acquaintanceship between a friendless abuse-victim reporter and a supervillain thief who wants revenge on said reporter's father and is the enemy of everyone else in said reporter's life. It's the most healthy because despite the difference and conflict, both always show a default level of respect to each other as intelligent fellow human beings. Yes, the relationships in this story are that screwed up.
Evangelion 303: Nearly all main characters suffer from some dysfunction. Shinji is half-orphan, his father has neglected him for nearly twenty years, and he had no expectatives for the future or even a regard for his own life until he met Asuka. Asuka is full with neuroses, insecurities, regrets and even though she seems strong she is very fragile and is prone to fall apart when it gets hard. Rei is an Emotionless Girl due to brain damage. Misato is getting constantly agnsty about her pilots and rebounding of a difficult previous relationship. Saburo is so full with tensions and frustrations that he almost is an emo...
Fallout: Equestria - Project Horizons takes this trope and runs with it. Blackjack is a none-too-smart bundle of self-loathing and guilt held together by chems, alcohol and Chronic Hero Syndrome. She's also guilty of rape and murder before the story even begins. P-21 has deep emotional scars and constantly stuggles with cognitive dissonance, particularly a repressed desire to kill Blackjack for raping him and killing his lover. Rampage is a nigh-invulnerable child-murdering psychopath with a split personality, who actually wants to be good and/or find a way to die. Lacunae is a live dumping ground for an entire Hive Mind's collective angst, prone to being the subject of Villain Overrides. Scotch Tape is a relatively sheltered Tagalong Kid who attracts psychological trauma like a magnet. The only (relatively) stable member of the main cast is Morning Glory, who quickly develops several problems of her own such as being betrayed and exiled from the Pegasi Enclave, branded, and later losing a wing.
In Fill the Moon, the cast all have some sort of horrible, crippling psychological issue. In order; Xemnas' impregnation of his own daughter using his surrogate younger brother as the father, Xigbar has dealt with a completely screwed-up lover, and is outright stated to have become a Knight Templar on his brothers' behalf during their time as Apprentices, and while Luxord, his lover isn't crazy, (which is even lampshaded by Xigbar), Xaldin has been through his entire life feeling alone, unloved, and "undeserving" of affection, Vexen, who worked himself beyond exhaustion being the Team Mom for the Apprentices, and is now dealing with the fact that they've grown up and don't need him anymore, Lexaeus is actually rather normal, but considering his girlfriend, Larxene, has been raped and abused in every single relationship she's had, and was horrifically abused by her ex-husband, who also raped her and is implied to have done far worse, he doesn't have to be crazy. Still more?
Axel is dealing with losing his best friend, Saix, to Xemnas, and pining for a lover who doesn't even exist outside of Sora yet, and has defected from the Organization to find him. Zexion, who is already canonically an orphan, is forced to impregnate his thirteen year old lover, who didn't even know what sex was until that catastrophe, is forced to watch her almost die due to the impregnation, and performs a Heroic Sacrifice in her name. Plus, he's implied to have even more trauma pre-Fill the Moon, so we're bound to see more on him. Marluxia is somewhat normal, but rather possessive. Dealing with Vexen could drive anyone crazy.
Demyx doesn't have it so good either. Being abused by his older brother for his entire life as a Somebody led to a serious case of PTSD, culminating in a massive mental breakdown before Xigbar managed to help him through it.
The worst of the lot is the little OC Nobody, Senayax. She was forced to eat her own grandfather to survive, she has a horrific demon living inside of her that expresses delight in the thought of raping her, she was impregnated by her lover under orders from her father, she almost died as a result, and committed suicide after Zexion performed a Heroic Sacrifice to save her.
To sum it up? Everyone in this fic is insane.
The First Class SOLDIERS in The Fifth Act. Cloud is actively plotting Hojo and Sephiroth's deaths due to the tragedies they inflicted apon him int the original timeline. Due to accidentally time-travelling he lost all his friends and sees the faces of long-dead one re-opening old emotional wounds. Angeal is the Team Dad of group and is hiding the fact that he's dying of a terminal illness. Sephiroth is disconnected from people and trying to get the approval of someone who despises him. It's telling that the resentful Drama Queen that is Genesis is the most stable one of the bunch.
Apparently, the entire Gensokyo 20XX series is built on this, especially later on, when we have Reimu who is age-regressed to a child (20XXI) and doped up with sedatives, along with being mentally conditioned (20XXII), later on developing mental health problems and becoming rather fatalistic on top of becoming an Ill Girl (20XXIV and 20XXV), Yukari,who is prison raped by the warden and goes insane (20XXII), Ran goes insane after a Near-Rape Experience and almost kills an age-regressed Reimu in a murder suicide, later on being raped and turned into bitch on top her of her other problems (she gets better), Chen is rendered blind, later on recovering from said blindness, crippled, and then she herself attempts suicide (20XXIV), babies being born and or dying almost soon after, if not being miscarried or stillborn, children growing up in chaos, and all other sorts of things terribly wrong. The kicker? Nothing can really be done about either of those, especially in the aftermath a of nuclear war.
HERZ: In the original series, most characters had some trauma they constantly got depressed or angsty about. In this story their traumas have mellowed down after twelve years -thankfully-, but Asuka still has anger management troubles, Shinji is still somewhat insecure
In How the Light Gets In, Team Arrow is portrayed as a dysfunctional mess at best, and a toxic trainwreck heading for a crash at worst. This doesn't stop them from doing their best to help people, but as Dean constantly points out, it hasn't stopped them from tearing into and dragging each other down over the years.
Gou's crusade against the Roidmudes was the only way to atone for his father's sins. His attempts to put his family back together backfires as it was revealed that Banno has never cared for his children, seeing them as research subjects. Chase's death was the last straw for Gou to make sure his father begs for his life as he kills him for good. He, like Micchy above who jokes around when depressed, going as far to bicker and troll one another. Gou has an obvious reason to be a Sad Clown not just for himself, but to everyone he cares about, like Mitsuzane and Samus.
Makoto and Kanon's mother died, and their father left them for unexplained reasons. Then came Takeru's dad, Ryu who raised the two siblings which in turn became friends with Takeru and Akari, whom they considered a family. Then, a freak accident sent the siblings to the Ganma world.
Alain gets thrown out of the Ganma world by Adel when he gets framed for their father's "death". Rather, Adonis gets regressed with a mortal body. It gets even grim when Adel finishes their father off, and Adonis dies in his youngest son's arms.
Heck, if you put the former Angelise Ikaruga Misurugiin the same room with both five isolated, and unbalanced individuals analogous to the above — Ange gets stripped off her royal title and exiled to a prison camp upon being exposed as a Norma (Alain), the bigotry and dislike towards the Norma changed her personality drastically given the fact sheisone (Gou), becoming an anti-social loner (Makoto), a hardened and pragmatic combatant (Samus) and in turn becoming more cynical and self-destructive, to the point of becoming morally grey (Mitsuzane).
The New Retcons: For starters, Elly Patterson lost her mind and thinks it's the 1980s in 2008. John refuses to do anything about it while she terrorizes her children and grandchildren, thinking Merrie and Robin are Michael and Liz, while completely denying that April is her daughter and throwing her out of the house. Meanwhile, Liz and Anthony are coping with realizing the person they married is not the person they thought they married and with Elizabeth cheating on Anthony, with her son not being his. Along with coping with an insane mother and a negligent father, Michael's trying to deal with taking in the displaced April, marital troubles, an autistic son, two miscarriages, finding out he's not John's biological son, and repressing his bisexuality. Then they all discover that Elly lost it cause she feared her bastard daughter showing up, a fear realized when their half-sister Claire appeared, causing John to attack Elly and Elly to end up in a mental hospital. Even then, Elly escaped twice, Michael's marriage is on the rocks when he let his workaholic tendencies get the better of him, Elizabeth and Anthony have setbacks in their own marriage, and John faces being known only as a crazy old man. Then, when Elly snaps out of it and attempts to make amends for her actions, she's murdered, and things went worse in Who Silenced Elly Patterson.
Final score: Elly's dead, murdered by her Psycho Lesbian best friend out of being a Woman Scorned, John cheated on her when she went in the mental hospital, married The Mistress (and oh, by the way, a woman that's April's age and had literally stolen from Elly) and died relatively young (younger than his father had, anyway) due to a bad diet, Deanna divorced Michael and got custody of the kids (and pretty much only speaks to April anymore), Elizabeth and Anthony are sticking together pretty much just for the kids, and April's estranged from most of her family. All would agree they're better off though.
All the major characters in the Paris BurningFan Verse are the personifications of major cities. Each one is subject to the trauma of their city's history and the stress of whatever hardships their current inhabitants face, compounded into one unusually long-lived person. Most of them are able to carry on, but needless to say they are not generally very happy people.
Power Rangers: Shuriken Force has all five rangers suffering from some kind of issue, and their mentor isn't much better. The Red Ranger is as green as can be, being painfully shy on top of it. The Blue Ranger tries to pick a fight and has shown misandristic tendencies. The Pink Ranger eggs Blue on, is defiant and snark at everything. The Yellow Ranger is immensely pessimistic and can't bring herself to care about much. The White Ranger at first seems normal, but her cheerfulness heavily is implied to mask a much frightening interior. And their mentor is so aloof that his own daughter can barely speak to him.
In A Prize for Three Empires, Carol Danvers considers the number of heroes with psychological issues, deep-seated traumas, troubled pasts and comes to the conclusion that super-teams are essentially dysfunctional families.
Like the Avengers and the X-Men before them, they had shown her that super-teams, at best, are little more than dysfunctional families.
All of the Avengers, and some of their friends and family, in the Second Chances Series have serious issues. According to Jarvis, the only members of the current or old Avengers teams that have never had therapy are Bruce Banner, Stephen Strange, and Natasha Romanov. And Jarvis feels the former two could seriously benefit from it. Besides the normal reasons some might need it, Vision sees a therapist to help understand being human, James Barnes has daily sessions to remove the Hydra triggers and cope with his time as the Winter Soldier, and Howard and Maria Stark needed to see a therapist to cope with having been murdered and resurrected.
Some time after Loki comes to Earth seeking asylum, Jarvis suggests he seek therapy as well, citing how many of the Avengers have needed it.
In The Wizard in the Shadows, its sequel and anthology of short stories, a good portion of the main cast are like this. Harry is borderline insane for most of the first fic - having immediately followed the Second Wizarding War by being trapped in Middle Earth for a good four years before the story even starts, with no real chance of return, and has been spending almost all that time conducting a one man war against every dark creature North of Gondor. Then he dies. And comes back. And Word of God has pretty much said 'You ain't seen nothing yet'.
Emrys Ap Derfel has a background that is arguably worse than Harry and was thrust into battle at the age of 14/15, after losing both his parents by the age of 8, suffering fairly epic racial discrimination from the moment he was born and having his sister sold to Isengard as a breeder (he being a smart boy, he very quickly figures out where the Uruk Hai come from).
His sister and sole parental figurewas kept as Wormtongue's personal sex toy in Isengard for four years (from the age of 19/20 onwards) in a literal case of So Beautiful It's a Curse, and molested by Uruk's for that time.
She is also painfully aware that she was only a week away from becoming a breeder anyway. Emrys is forcibly confronted with the fact that he was nearly uncle to an Uruk child and when asked, is not sure what he would have done. And he's sixteen/seventeen at this point.
Sirius spent a good six or seven years in Saruman's dungeons being tortured mentally and physically.
Zenith, Darkness, Reverie, a Death Note fanfiction, combines an insolent shinigami with an exceptionally bleak outlook on life, smothering maternal figures, a mentally shattered teenager (two diagnosed personality disorders, denial of every aspect of her existence, peculiarly elevated intelligence, and severe psychopathic tendencies) with a mass murderer intent on psychologically and physically destroying the entire cast. Said murderer is an alternate personality of the teenager. Such fun.