Avas Demon Poor Ava is constantly tormented and possessed by the Demon Wrathia, all in an attempt to make her commit suicide. It seems like if anything can go wrong for Ava, it will go wrong.
Unsounded: Rector Duane Adelier's a goofy, kindhearted family man who defends his daughter's life in vain, watches her die, himself dies, is reanimated with currently unknown means, and then consciously endures his own decay into a ravenous corpse-construct over a period of six years.
Hanna Is Not a Boy's Name features Hanna after the fact, though Tessa featured a time line of his life where he was a happy go lucky kid, something absolutely terrible happened that brought him down to a severe depression/dark stage but then he got back up again. Some things he still refuses to talk about, but has returned to a somewhat normal cutie goof ball of hyper activity.
Homestuck. Jade starts out as the only one who has an awakened dream self. Whenever she falls asleep, she has the greatest fun in the world flying around Prospit's moon and hitting her dead grandpa in the face with her dream robot. Then, when the Sovereign Slayer/Jack Noir destroys Prospit and drops its moon on Skaia, dream!Jade has to save dream!John from the imminent collision. Realizing he just will not wake up, she throws him out of the way, sacrificing herself. This means as soon as she wakes up, no more flight around Prospit's moon, no more precognition, and worse, since dream!John just woke up, she just missed her opportunity to fly with him, something she wanted to do from at least a young age. After waking up, she immediately fell asleep again due to her narcolepsy, causing her to have an incredibly traumatic dream with Lovecraftian horrorterrors. Poor Jade.
But Jade is far from broken. Dream Jade, on the other hand, is very much so, having been brought back as Jade's sprite, which gives her First Guardian powers AND forces her to have to see the green sun at all times, plus the trauma of being brought back to life is simply too much for her and she ends up being inconsolable and useless since all she can do is cry and go on about how good things were before she was brought back to life. It doesn't help that the first thing Jade asked Dream Jade to do was go confront Jack Noir.
Aradia, good lord. The only thing keeping this from being a devastating Player Punch is that the breaking happens in flashback, so we first meet her as the apathetic shell who brings everyone down without trying. Only later do we see her as the impetuousGenki Girl whose main goal seemed to protect (and then avenge) her much-bullied friend. Something terrible must have happened to her, like her boyfriend being mind-controlled into killing her. She eventually gets put back together.
Karkat, while his status as 'cutie' is arguable, has certainly been through enough to make it onto this list. Near the beginning of Hivebent his lusus was crushed, and he checked his laptop to discover that his good friend had died horribly, partly at his own fault. (Even though he did get better, it was still pretty traumatic for poor Karkat.) Then a teammate of his is murdered and we get to watch him desperately try to revive her. Shortly afterwards, another friend goes berserk, brutally murders two more friends and carries their remains around the building, all while threatening Karkat's life through other people's chat handles. Oh, and most of these events take place on a deserted asteroid where he and his team have been hiding since their entire home planet was destroyed.
Like Karkat, it could be argued as to whether or not Gamzee is cute. What is certain however, is that he is pretty severely broken by the demolition of his faith and the inherent psychological issues that come with being purple-blooded. Needless to say, it sends him off the deep end
Probably the best example, if a retroactive one, is Damara. She used to be a totally sweet, shy girl who was head over heels for Rufioh. However, when she found out that he was cheating on her - and when Meenah deliberately twisted the knife just to see if she could make her snap - she went full Yandere and transformed into a psychotic, cynical Jerkass.
Surprisingly enough the once confident and strong Terezi became possibly one of the most broken characters in the comic thus far(other than Damara, that is.)She's about a million times more the self loathing mess than Karkat was and is stuck in what is definitely a very unhealthy and abusive relationship that prompted her to change the thing about herself she was most confident about.
Early webcomic Roomies! is probably responsible for this, when it broke Joyce Brown by showing porn to her (yeah). Years later in It's Walky, Joyce isn't so innocent, but thanks to some convenient Applied Phlebotinum, she can't remember the incident and still thinks it was something absolutely horrible.
And then proves to be a Cutie who snaps, rather than breaking...
In Shortpacked! David Willis turns this trope into a huge subversion salad with Amber and Mike's relationship. It starts when Amber blackmails Mike to pose as his boyfriend by threatening to expose his Dark Secret. Mike sets up an elaborated scheme to break the cutie to get free, giving Amber several blows in the process - but stops short from finalizing it because he's come to appreciate certain aspects of the pseudo relationship.
He finally ends up giving the final blow when Amber starts pushing him towards marriage - now because he's genuinely fallen for her and thinks she deserves someone better. It comes a little late however, since she's aware of his affection and sees right through the ruse.
Grace from El Goonish Shive, during the most Cerebus-y arc, "Painted Black". She gets unbroken. Bonus points for Ellen during "Sister", if we can really call her The Cutie.
Actually, you could say Ellen started off broken by being transformed into a cutie in the first place, making it "Heal the cutie" instead.
Justin was very broken before Elliot befriended him ( he's improved but still has plenty of problems), Tedd's Friendless Background and implied family and gender identity issues, which has left him very insecure, and Susan who is ridiculously jaded by the time we first meet her thanks to catching her father cheating on her mother, being raised to hate men, pressured to be a lesbian (she isn't), and forced to kill a very human-like demon by decapitation at a young age.
Grace gets broken again when she starts learning about history. Interestingly, it's not the fact that it happened, but the fact that it could have happened again with Damien in the main role.
In Ronin Galaxy, Taylor seems to be the cutest character who first this role the best. In chapter one, she is a waitress trying to escape servitude and other unspeakable things by the Yakuza. Unfortunately for her, it ends up that Cecil and Giancarlo make her their maid anyways.
The Order of the Stick has a point where Haley snaps after all the treasure they gathered from a dragons horde went up in flames, following an incident involving explosives in the tavern they were staying at. She spends the next several dozen strips unable to speak in anything except gibberish, and slowly developing a host of ghostly subconscious voices counseling her to confess her secrets. Still quite funny how it plays out.
Coincidentally, this same arc culminated in Nale telling Elan that Haley is Evil All Along and The Mole and (with the help of some Brainwashing) drives Elan into briefly snapping and nearly killing Haley before Haley manages to break out of her gibberish-speak by finally confessing to Elan that she loves him.
Elan also endures this during the Therkla arc. The entire story of that (once the character comes in) is Elan being punished for the na´ve loyalty he shows to Haley, up to and including Therkla dying rather than living without Elan. What's surprising is how he comes out.
Poor Elan gets broken again in #889, when he's forced to admit that his dream of his Dysfunctional Family happily getting back together is childish and implausible in order to break himself and his friends out of one of Girard's illusions.
Sluggy Freelance has a 6-month storyline apparently dedicated to accomplishing this.
It happens to Fumbles in Goblins. After he wounds a little girl by accident, he decides to make things right, and ceases to be the comic relief he was earlier. Later, it gets worse, when Dellyn Goblinslayer forces him through a brutal torture session, climaxing in carving the word "Monster" on his forehead. The next time he is seen, Fumbles is shown to be mentally broken and mute.
May have happened to Saves-a-Fox when her entire worldview is ripped out from under her by Dies-Horribly explaining the true, fatalistic implications of her killing the fox. Cue Heroic BSOD. Not a good time for it either.
Luna starts out like this in Dominic Deegan, and gradually gets better. Currently, she's unbreakable, even though almost every villainous character who knows about her past has tried to rebreak her. Nimmel becomes this for quite some time, but, again, gets better. Actually, almost every major character in DD has come fairly close to breaking but has either been able to get better or are able to pull themselves together at the very brink.
City of Reality has this occur to a main cast member, the saccharinely idealistic Todo, who suffers a hard jolt of reality when confronted with the common cruelty of the World of Magic.
In Megatokyo there was a comic (a few actually) where Miho was playing head games with a drunken Largo. Piro walked over and saw what she did. Cue his Berserk Button . Afterwards she cried. 
Faen in Drowtales has experienced multiple Break the Cutie moments in succession. What makes it worse is that she's The Empath and incredibly sensitive to the most minute emotions.
Ariel goes through this too. Kyo'nne actually has it pretty easy, although being rejected one too many times by Shan'naal certainly hurt. Naal'suul is a pre-broken cutie, as is Kiel'ndia (although it's more obvious with Naal). It would probably be easier to list characters from Drowtales who haven't been through a round of Break The Cutie at some point.
That's easy: Melodia, Ariel's "pet" Feral. Though after what happened to the last cat...
Nana's Everyday Life, which takes one of the characters from Elfen Lied (see very first example on this page) and does it to her some more. Make that a lot more.
Happens to a degree with Laura in Collar 6, as she realizes just how seriously she's getting involved in BDSM.
In The Inexplicable Adventures of Bob!, adorable baby peanut butter monsters reach maturity in a single month. Molly spent her month being raised in a loving home environment by Bob, and is now very cheerful. Her sister Golly spent her month being treated as a lab animal and never knowing even a moment's love. She is... markedly less cheerful than her sister.
Anna in Sire. She attempts in vain to make a personal connection to anyone, and ends up slaughtering the only person she successfully can befriend, before being arrested, locked in an insane asylum, escaping, fleeing the country, and being abandoned to a dubious fate. The worst part? Her alternate personality is the one doing it all, so there's no way to escape.
Subverted in The Challenges of Zona in that yes, Mentl turned Ginsha into a stuttering alcoholic who's psychosexually addicted to him, but he did it in self defense and unintentionally. He's not even aware that he has.
Tekno in Sonic The Comic Online. For years she's been conceived as a gentle, Teen Genius. Recently however she has gone through a mental breakdown due to her being away from technology for months on end, even murdering a character, which she would have never done in her normal state of mind. She also attacked him physically, which is rare since she usually doesn't attack directly herself but use technology to do the job. She's also never used lethal force. However it's implied she did horrid things in her past.
This trope was cleverly dodged in Frog Raccoon Strawberry after Marco failed her attempt to confess her love to Josh. But just before she could have a breakdown, Josh showed up and serenaded her.
Charlie in Khaos Komix was a sweet, innocent naive young cutie, until Natalie Geln and her friends ambush her and her best friend Tom, knock Tom out with a rock, threaten to rape him (He was a FTM transsexual) and cut her hair off before going on a long speech about how she doesn't want any freaks in 'her' school, and they should 'do the world a favour and kill themselves'. Granted, Natalie was never a nice character, but after this, many fans were baying for her blood. Which they never get.
Kokkan in "A Broken Winter". First he's tortured for information regarding his missing terrorist father, then he's sent to be executed at Irihi but spared and placed on the clean-up crew for aforementioned executions. He's spared again when the other members of his crew are executed in front of him. His breaking point comes when he's forced, under threats of torture and violence, to kill an infant. He's seven. It's no wonder the poor child grows up with a crippling case of PTSD.
Invoked in Girl Genius, when Doctors Mittlemind and Mezzasalma resolve to do this to poor Theo, here. Made worse (or funnier) by the fact that they treat it like a game.
"Webcomic/Threading": Deirdre gets pushed around by basically everyone who's shown up so far.
Heart Core: Amethyst Lashiec, once a young, payful, kind, loving and naive girl despite being a princess of Asgard and the daughter of a mighty demon. Then her dad gleefully put her through a painful ritual against her will, costing her much of her powers while leaving her with permanent scars. She was not happy about this...