Avon, during the show's fourth and final series. The final straw comes with Blake's (apparent) betrayal. Considering what he's been through, the only wonder is it didn't happen sooner.
Also Travis, who starts off as cruel and obsessive, but rational, and ends up as a homicidal maniac who sells humanity out to alien invaders just to get a chance to kill Blake.
Breaking Bad: Walter White displays this at the end of "Crawl Space" laughing maniacally after realising that Gus Fring is going to kill his family and Skyler gave his money to Ted Baneke. Many feel that this is the point where Walter White completely died and only his alter ego, Heisenberg, remained.
Buffy the Vampire Slayer: Several examples. Faith, portrayed as not the most stable of individuals to begin with, begins by trying to rape Xander to death and it goes downhill from there. After Tara's death Willow thinks nothing of killing her friends if they get in her way of vengeance, before deciding that destroying the world would be a Mercy Kill. Even Buffy herself is prone to this: the stress of each season finale seems to bring out the worst in her (such as trying to drive her friends away, fleeing from Sunnydale, thinking she can't let anyone close out of fear of getting hurt, then It Gets Worse, oh boy does it ever get worse. Happens to Whistler and Angel bluntly tells him that he believes that being separated from The Multiverse and the Powers since the Seed was destroyed have made him crazy. And he may have a point.
Dexter wrestles with Sanity Slippage during the second season of the series.
Mrs Tishell on Doc Martin. Her crush on Martin went from "she's an odd one", to "what're those pills she's taking?", to Stalker Shrine, and finally a full blown psychotic break, which was a bad interaction between two drugs she was taking.
The Master has always been rather unstable, try as he might to hide it under his veneer of a Magnificent Bastard. However, this takes a turn for the worse in NuWho, when he complains of a persistent drumming in his head, getting even worse after returning from the dead in the Specials Season. The main question is whether or not the drumming is real, making the Master even crazier (It is, turning out to be part of a Gambit Roulette by the Lord President of Gallifrey to free Gallifrey from the time-locked Time War). With this revealed, fans speculated on how bad the drumming was during the Master's previous incarnations.
Whatever she did to get out of the pocket dimension apparently pushed Missy completely over the edge into a perpetually cheerful, self admitted "bananas" borderline Cloud Cuckoo Lander, who skips around, breaks into song apropos of nothing and randomly switches between an English and Scottish accent. She's also the most sadistic and pointlessly cruel incarnation yet.
Hell, the Doctor himself gets this sometimes! The Eighth Doctor undergoes this twice in "Minuet in Hell" and "Zagreus", The Tenth Doctor has one in "The Waters of Mars" that temporarily turns him into the Time Lord Victorious, and the final three episodes of Series 9 see the Doctor Freak-Out and undergo a Protagonist Journey to Villain — becoming a Woobie, Destroyer of Worlds for a while — when he proves unable to handle his anguish and rage over Clara's death. It's telling he needs Mind Rape - delivered unto him by no less than Clara herself (though he does this voluntarily) to help him return to his best self in the end. The revival series has made it pretty clear that without companions to keep him grounded, the Doctor would likely undergo these slippages all the time. Witness this exchange from "The Runaway Bride":
The Doctor I don't need anyone. Donna Noble Yes, you do. Because sometimes, I think you need someone to stop you.
The main protagonist, John Crichton, goes increasingly insane over the course of the second season. Partly it's due to having an Enemy Within surgically implanted into his head, but much of it is simply accumulated trauma. He gets better in the third and fourth seasons, but never completely.
Crais becomes increasingly unhinged over the first season, as his initially not-very-rational quest for vengeance over the (accidental) death of his brother gradually destroys his previous ideals and connections to others. He gets better at the end of the season, although it makes him even less predictable.
Stark, when initially introduced, is not completely stable but vastly exaggerating his madness to confuse his jailers. However, after being executed by disintegration and having to create a new physical body for himself in the second season, Stark returns with a distinct tendency towards panicky meltdowns and screaming fits; season three only makes things worse when his lover Zhaan dies, leaving him swinging wildly between near-catatonic despondency, psychotic rage, obsessive spells, and explosive paranoia - and that's when he isn't just twitching and mumbling to himself.
Firefly: One of the tie-in bits of promotional material for the movie Serenity is a short series of films called the "R. Tam Sessions," which depict River's traumatic time at the Academy. It starts off with River being a happy, eager girl who really wants to learn and push herself to her limits, but as the series progresses, she is shown slipping into madness due to the Academy's experiments. The series ends with River killing the man who has been interviewing her by shoving a pen through his throat (who, for those of you playing at home, was played by Joss Whedon).
From Dusk Till Dawn: Richie Gecko used to be a highly intelligent, methodical and dependable bank robber. However while his brother Seth was in prison he became more and more unhinged, and it only gets worse as the series starts.
Ramsay Bolton. Not that he was ever stable to begin with, but according to Word of Godhe is unable to process that he is losing from the moment where the Knights of the Vale arrived in "The Battle of the Bastards" as he is used to being in control and get his way every time. He remains confident that his cunning and tricks will get him out somehow. When his army is absolutely destroyed by the Vale knights, he waves it off as nothing saying he still has Winterfell and can withstand a siege, never mind that the Arryn forces outnumber him and are prepared to besiege him. When he is being beaten by Jon, he thinks he will be spared because of Stark honor, without realizing Jon spared him so Sansa, Jon's sister, could be the one to deal the final blow. Even when he is chained up inside the dogs kennel, he still thinks he will break out of this one for some reason until they are literally on top of him. This echoes the last days of Adolf Hitler, who acted much the same way regarding failure.
Cersei is much more calm, collected, and rational in the first season than she is in the second. Her loss of Jaime and Joffrey's increasingly out of control attitude, complemented by her father's suddenly dismissive attitude towards her don't seem to be doing well for her mental faculties. Her despair over her daughter Myrcella's impending Arranged Marriage can't help, either. She comes within inches of poisoning Tommen during "Blackwater" when she thinks that Stannis is about to break down the door. In "The Lion and the Rose", she's outright screaming with rage at Tyrion, as she believes he poisoned Joffrey, and by Season 5 she's noticeably resorting to alcohol continuously, further worsening her condition. By the end of Season 6, she's clearly gone off the deep end. She not only blows up the Sept of Baelor and dozens of nobles within, including her uncle Kevan and three of the four Tyrells, but she doesn't care at all that Tommen kills himself in the aftermath. She still believes she can create a great dynasty with her children dead and surrounded by enemies. In the Season 7 premiere, Jaime actually calls her out on this last point.
Cersei: I understand whoever wins could launch a dynasty that lasts a thousand years.
Jaime: A dynasty for whom? Our children are dead. We're the last of us.
Specifically, Will has extreme empathy that he uses to solve murders, and encephalitis. Will's psychiatrist is a serial killer. Will's psychiatrist knows he has encephalitis, but as it worsens convinces Will that his hallucinations, sleepwalking, lost time and seizures have no physical cause. Will's psychiatrist encourages him to think that killing a different serial killer in the first episode felt good, manipulates him into shooting another serial killer while hallucinating the previous one, and frames him for multiple murders. Will actually copes remarkably well, given the circumstances.
Heroes: During Volume 4, Sylar's already-unstable psyche takes a dramatic turn for the worse when he acquires the ability to shapeshift and starts to lose his sense of identity after using it repeatedly.
House: In late Season 5, House's mental state quickly begins to deteriorate into hallucinations of Amber and delusions of a romantic relationship with Cuddy. House agrees to be voluntarily admitted to Mayfield Psychiatric Hospital.
It's Always Sunny in Philadelphia: all of the characters become increasingly unhinged throughout the series, with Dennis deserving special mention. While he starts out as the Only Sane Man of the group, he is slowly revealed to hide anxiety, feelings of emptiness, and sociopathic tendencies beneath his Mask of Sanity. As he grows older, he finds it harder and harder to keep these issues under wraps, and he is eventually diagnozed with borderline personaloty disorder and prescribed medication.
M*A*S*H: Although it's never shown onscreen, during the finale, after witnessing a traumatic event, Hawkeye begins to not-so-subtly show signs of a breakdown. Like being convinced one of the anesthesiologists was attempting to suffocate his patient with the gas mask, or crashing a jeep through the mess tent. Little indications of a stressful work environment.
In "Bananas, Crackers and Nuts," Hawkeye fakes sanity slippage in order to weasel some R&R in Tokyo.
Melrose Place: Kimberly, big time. Kimberly initially appeared as a wholesome and ethical doctor friend of Michael's, before beginning an affair with him. Then, a car crash in season 2 nearly killed her. From then on, she became more unhinged, with her acts running the gamut from running down Michael with her car, framing Sydney for the hit-and-run (and actually getting away with it), kidnapping Jo's baby, hiring people to assault Matt to prevent him from leaking revealing test results about her sanity, developing a multiple personality disorder, institutionalizing Peter whilst under one of her personalities, and most strikingly...blowing up an apartment building with people who didn't really do anything to her. By season 5, Kimberly began to regain her sanity...only to die from an unexpected aneurysm. Soap opera, indeed.
Merlin: Morgana starts out pretty sane, but after she goes evil she starts going off the edge. She is encouraged by her sister to isolate herself from everyone nonmagical, and when her sister dies becomes obsessive about gaining her revenge. During the Time Skip between seasons four and five, she was locked at the bottom of a well and didn't see the light for two years, which has damaged her beyond all repair. Even Mordred goes from idolizing her to being frightened by how unhinged she is and stabs her in the back when she tries to kill Arthur.
Simon Bellamy from this British sci-fi series is a painfully shy and intense introvert (and convicted arsonist) who has been bullied and ignored all his life; it's hinted from the onset that he's teetering on the verge of mental breakdown. As the series continues he accidentally becomes party to murder, is magically imbued with the power of invisibility (which, awesome as it sounds, greatly heightens his growing sense of alienation and his unhealthy tendencies towards voyeurism) and inadvertently causes the death of Sally - a woman he was starting to fall in love with. During the final episode of Season 1, the full extent of his sanity slippage is magnificently depicted as he casually munches on some left-over pizza while gazing serenely at Sally's corpse, which he has propped up in a large freezer. He got better though.
Series one also shows Sally the probation worker go through a (arguably milder) version of this. Tony, the previous probation worker was her fiance and she spends the whole season trying to prove that the gang is behind his mysterious disappearance. (and she's right about it), though she spends most her time stalking the misfits and by the end became completely obsessed by this.
Praline: I am not a loony! Why should I be tarred with the epithet 'loony' merely because I have a pet halibut? I've heard tell that Sir Gerald Nabarrao has a pet prawn called Simon, and you wouldn't call Sir Gerald a loony, would you? Furthermore, Dawn Palethorpe the lady show jumper has a clam called Sir Stafford after the late chancellor. Alan Bullock has two pikes both called Norman and the late, great Marcel Proust had an 'addock. If you're calling the author of A La Recherché du Temps Perdu a loony, I shall have to ask you to step outside!
Invasion of the Neptune Men, in which Tom is so unnerved by the use of stock footage of World War II bombing runs (as indicated by the Hitler building) that he comes totally unglued.
"Manos" The Hands of Fate, which was so bad that Joel, the most laid-back riffer on the show, eventually cried out "Do something!"
Newhart: Dick suffers a variant on this in one episode thanks to sleep-deprivation.
Orange Is the New Black: Implied; we still have no idea whether the chicken or the voice Piper heard in SHU are real. In the Season 2 premiere, Piper starts talking about her painting made of rotten egg yolks, Thirsty Bird, among various other incidents during Season 2. Vaults straight into Nightmare Fuel. Though, it is shown in Season 3 that the chicken is real and gets in and out of the prison through a hole in the fence.
Oz: Many a character undergoes this, especially Beecher in season 2 after taking a level in badass. He gets better.
Person of Interest: In mid-season four, after a season and a half of Sanity Strengthening, Root was, while still a bit of a lunatic, much more stable and caring about people's lives. After Shaw's Uncertain Doom, however, she slowly loses it, reverting to her more psychopathic tendencies and being more than willing to brutally torture people for information. She even decides to kill an innocent woman in order to protect Finch when there were several other options available to her.
Since this is a post-apocalyptic setting, it's a given that a lot of characters have experienced this.
It's suggested by several characters involved in the militia (like Jeremy Baker in episode 3) that Monroe's been suffering from this somewhat ever since Miles left him. episode 10 is when this trope becomes obvious for Bass Monroe. In episode 15, Emma Bennett, Monroe's high-school girlfriend, dies before she could tell him where to find their son, and that had to have added to his insanity. The first season finale reveals that it wasn't just Miles leaving him. It was also the fact that Miles tried to kill him while he was sleeping and leaving without an explanation. The fact that Monroe has a borderline erotic fixation on Miles did not help matters.
Ray Kinsey in episode 4 fell victim to this as a result of watching his daughter die.
Drexel in episode 6 has this going on, but he is a sociopathic drug lord who has likely become addicted to his own heroin.
In episode 12, Randall may have undergone one triggered by the death of his son, resulting in him co-opting the Mathesons' research to produce a superweapon—initially to end the war in Afghanistan but ultimately to destroy the world order and start anew.
Rachel Matheson has been undergoing this since the blackout, as revealed in a flashback in episode 19. It only got worse after the death of her son Danny in episode 11.
St. Elsewhere: Peter White suffers from this during his three seasons on the show, culminating in his wife throwing him out and his subsequent spiraling into drug and alcohol abuse, sex addiction, a DUI arrest, and ultimately his rape of several nurses, one of whom eventually shoots and kills him.
The Sifl and Olly Show: Olly is an odd case in that his sanity is only shown to ever slip during the "Precious Roy" segments, which his psychiatrist describes as "manic-compulsive salesmanship". Olly goes further and further off the rails each time the segment comes around, but as soon as the next segment comes, he's back to normal.
Smallville: Major Zod of the 9th Season started out bad and only got worse from there.
Antagonist Dukat. He starts off as a scheming, megalomaniac, yet charming villain, gets gradually worse, especially in season six, until it all finally boils down to a psychotic Motive Rant that is declared in-universe to be a Moral Event Horizon: "I should've killed every last one of them! I should've turned their planet into a graveyard the likes of which the galaxy had never seen! I should've killed them all."
Faked—probably—by Sisko in For the Uniform. He even slips into the Joran voice.
In "The Omega Glory," after Tracey learns all his evil deeds are All for Nothing, you can see him snap at the news and proceeds to threaten Kirk for weapons for a pointless fight on a planet they by now can leave any time.
The very next episode, "The Ultimate Computer", when Richard Daystrom attempts to convince the M-5 Computer to stand down after it has killed numerous crewmen on various ships, the fact that his creation, and by extension him, has done this weighs so heavily on him that he snaps, first ranting about how his colleagues laughed at him and took his work to make their own careers, then snapping at Kirk and declaring the two of them invincible at how powerful its become. Spock is forced to Nerve Pinch him before he brings harm to Kirk.
Season seven does this to Sam across most of the season, culminating in The Born-Again Identity.
Also happens, during season seven, to Cas, when he absorbs all of the souls of Purgatory and later on, when he is possessed by the Leviathan.
After getting the Mark of Cain, Dean has become unstable and enjoys killing versus before when he would show remorse about it, and feels absolutely no regret about it. He even tells Sam that they do things his way and it's not a team anymore but a dictatorship. Culminates in him attacking Gadreel while growling at the end of "Stairway to Heaven."
Has some contestants who seem to have gone a little nuts during their time in the wilderness:
Survivor: Amazon had Matthew who freaked out his teammates by beginning to very meticulously sharpen the machete every day while staring creepily out into space, and Butch who suddenly gained a firewood fetish and forced his teammates to stockpile all the wood they could find which ended up burning their camp down when said massive woodpile caught fire. Even Matt thought that was a little messed up.
Panama had Shane who made the genius decision to quit a 3-pack smoking habit literally hours before getting on Survivor. Cue him yelling at his tribemates every single opportunity he got, treating a hunk of wood like a Blackberry, and looking more and more like a crazed animal the longer he stayed out there.
Heroes vs. Villains saw Russell get hit hard with this trope. Parvati thought it would be just peachy to keep a hidden immunity idol from him just to see him squirm. Once she handed them to Jerri and Sandra to nullify all of the Heroes' votes for the former, the Disaster Dominoes began. For both sides.
Only Sane Man Alex fell into this whenever he really got over the edge. In an early episode, Alex was given some pills he thought that were aspirin, but these were actually "uppers". Their effects are practically immediate, with Alex talking very fast and singing "Bye-Bye, Love" after waving goodbye to the rest of the gang when he goes to see Mr. McKenzie. Later he ripped Louie's cage with his bare hands and threatened Louie because of his refusal to listen to him about re-hiring Bobby.
Teen Wolf: In season 2 Lydia begins hallucinating after being attacked by Peter, and in the first episode escapes from the hospital and runs naked into the woods. However it's ambiguous how much of her 'insanity' is actually her going crazy and how much is Peter's ghost gaslighting her. It seems to be more the latter, as Peter is shown to be able to turn her hallucinations on and off, as well as having some degree of Mind Control over her.
Later comes up in season 3: after sacrificing themselves to the nemeton, Scott, Allison, and Stiles all begin exhibiting this. Scott can't control his werewolf transformations, Allison hallucinates her dead aunt, and Stiles is experiencing dreams and hallucinations so vivid (along with occasionally losing the ability to read) that he's no longer able to tell when he's awake.