In The Christmas That Almost Wasn't, after the Big Bad Prune buys Prim's Department Store and basically steals the money Santa earned to pay for toys he damaged, deliberately screwing Santa over from paying the rent he owes to Prune, Santa and Whipple stroll listlessly down the street, hoping desperately for a miracle...
Malcolm Reynolds lost all idealism, along with any faith in God, at the battle of Serenity Valley which meant that the Independent Planets will be unified with and ruled over by the morally ambiguous if not downright evil Alliance.
River Tam's dialogue indicates that she has no hope of ever being "normal" again. At one point, she even rails against the drugs Simon is giving her, saying that she hates being able to think clearly because she knows she'll just slip back into madness sooner or later. You can actually see the very moment River breaks in the R. Tam Sessions, in the third video where the "counselor" tells River that her brother "is very busy". She stops, silently nods, then whispers "Yes...." and starts crying. (crack)
Halfway through season four of Battlestar Galactica, the fleet discovers that Earth has been destroyed, causing borderline cases of this for many characters, and a full-blown case for both Dualla and Number Three.
Series 3 of Torchwood has multiple Despair Event Horizons - as you would expect from a plot that involves the governments of the world caving into an alien race's demands for 10% of the Earth's children. Obstructive BureaucratAnti-Villain Frobisher is told by the Prime Minister that, for PR purposes, his daughters will be among the 10% given to the aliens - so Frobisher takes a gun home and kills his wife and children, before turning it on himself. Then Jack, who has just had to watch his boyfriend die pointlessly, realises that there's a way to defeat the aliens after all... but it will mean killing a child. And the only child close to hand is his own grandson, who he is forced to murder in front of his screaming daughter. Safe to say, Jack doesn't hang around on Earth for very long after that.
Dead Set. Space agrees to unlock the Diary Room so Kelly can make a futile attempt to fight her way out only after seeing a zombie Pippa (his former girlfriend) hammering on the glass outside.
Joseph Bede crosses it in the final episode of The Shadow Line. His wife, whose Alzheimer's treatment he intends to pay for with the proceeds from his drug deal, tries to kill herself and is hospitalised. This prompts him to leave his gun behind when meeting Jay, who he already know intends to betray him, which leads to his death.
The AR version of Kamen Rider Amazon suffers this in his arc of Kamen Rider Decade when the young boy he befriended turned against him, stealing the Gigi Bracelet in an attempt to rejoin Dai Shocker. It gets worse when his last hideaway is ransacked and he's close to taking Yuusuke's offer to join them in world hopping.
The DEH rests at the core of Kamen Rider Wizard. Individuals called Gates possess an inner demon called a Phantom, which is held in check by their will to live; crossing the Horizon allows the Phantom to break free into the real world, killing their host in the process. Thus the Phantoms that already manifested try to expand their numbers by forcing Gates past the Horizon while Wizard fights them off and tries to be the last hope for their victims. However, if a Gate comes close to the Horizon but regains their hope through Heroic Willpower, they gain the ability to tap into their Phantom's magical powers and can (with training) become magic-users like Wizard; this happens to three individuals over the course of the series, all of whom become Kamen Rider Mage.
Warehouse 13. Quite a few plots are centered around crossing the line, most notably in Reset, where H.G. Wells turns on the agents to put the world into a new ice age, effectively resetting human life. She is stopped at the last second by Myka Bering.
Dean Winchester crosses the line in the season 2 finale of Supernatural, after his brother Sam dies in his arms. He ends up selling his soul to bring him back, and after a year is sent to hell. This leads to him crossing the line again; after 30 years of torture he breaks and takes up on Alistair's offer to torture other souls. Ironically, in the Season 5 finale, Dean is the only one of the main characters not to lose hope after Lucifer possesses Sam. Dean still had enough faith in his brother to wade into an apocalyptic showdown and try to get through to Sam. Although to be fair, he made it clear that he had absolutely nothing to lose at that point. In Season 9, he is so wracked with guilt over everything that has gone wrong as a result of his initially ill-thought decision to have an angel possess Sam that he trashes the room he's in, takes off as soon as he ascertains that Sam is alright, then takes on the deadly Mark of Cain- while literally cutting off Cain mid-sentence when he tries to explain the consequences of the Mark. He literally no longer cared what happened to him at that point.
Sam has also had several of these over the course of the series, especially times when Dean has died/Sam has believed him to be dead. In particular, after Dean went to Hell at the end of Season 3, Sam drank himself into a near-stupor and was overtly suicidal until Ruby propelled him into a revenge mission. When Dean disappeared at the end of Season 7, Sam "imploded," stopped any kind of hunting, and drove aimlessly for months- which was quite unfortunate, as Dean was in fact alive and could have really used the help. Between Sam's shame at not saving Dean (combined with other mistakes he'd made over the course of the series) and being worn down and ill by the Trials in Season 8, he seemed to hit that point again and by the end of Season 8 was entirely okay with dying to finish the Trials (after taking them on initially because he wanted to live and thought Dean would use the Trials as an elaborate suicide). Then again in "First Born" in Season 9, when Sam discovered that Dean tricked him into accepting an angel possession to save his life, resulting in Kevin's death at Sam's hands, he became suicidal again and practically urged Cas to kill him in a slim-chance attempt to track the aforementioned angel, as he felt he never should have survived in the first place.
A Serial Killer uses this as part of his MO in the Cold Case episode "The Road". A Serial Killer known as John Smith would capture women, seal them in a room in the middle of nowhere, and subject them to Mind Rape. Once they lost the will to live, which he would prove by standing aside and giving them the option of walking out, he would seal them in completely and let them starve. He comments that once someone gives up hope, dying is just a formality. The fact that his last victim didn't cross the Despair Event Horizon really rattles him.
JAG: In "King of the Fleas", Roscoe Martin tells the story how had as a POW during The Vietnam War been tricked into revealing the names of the pilot inmates at the POW Camp, and was then forced to watch as the Vietcong executed the rest of them, which had plagued him ever since. Subverted in that after he had killed the camp commander in the present-day (1997-98), he regained hope again, as evidently seen in "The Martin Baker Fan Club".
Horatio Hornblower: Young midshipman Archie Kennedy loses his will to live in "The Duchess and the Devil", the third episode of the miniseries. He was horribly tormented and abused by a sadistic bully of a midshipman. He got lost at sea during a stealth raid (his tormentor tried to murder him), and he ended up in Spanish jail. His escape attempts earned him Cold-Blooded Torture in a Punishment Box. When, by some incredible chance, his friend and fellow middie Horatio appears in the very same cell, it brings back memories of his tormentor and he also feels that his life is one big screw-up, compared to the heroic and successful Horatio. When Hornblower insists that they will all try to escape together once he gets better, he doesn't want to. He decides to starve himself to death so that Horatio and his men needn't deal with him. When Horatio realizes what's happening at the last moment, Archie refuses his help. Horatio convinces him that he needs him. He eventually gets better, but it was hard and painful process.
House of Cards: Congressman Peter Russo, who's running for Governor of Pennsylvania, has this after the drunken radio interview he gave after Underwood set him up to fall off the wagon with an ex-call girl who owed him favors. He hid from the press and disappeared for a few days, leaving his campaign in limbo, before being found by Underwood who plied him with more liquor and murdered him, making it look like a suicide.
The Walking Dead: Rick gets two of these. The first after he's forced to kill his best friend in the season finale and the second after Lori dies in Season 3.