The Wall by Pink Floyd is just one colossal DEH; the entire album is about a rock star who is constantly hurt within his life, and the mental "Wall" he builds between himself and society. Summed up in the aptly titled 'Goodbye Cruel World', as Pink is having a mental breakdown and going catatonic:
Goodbye cruel world,
I'm leaving you today.
Goodbye, goodbye, goodbye.
Goodbye, all you people,
There's nothing you can say
To make me change my mind.
Rammstein's song "Wo Bist Du" has the narrator crossing the Despair Event Horizon after the death of loved one.
Jethro Tull's song "Locomotive Breath" is about a man who has just crossed the Despair Event Horizon.
Supertramp enjoys these. Lord, is it mine?, Rudy (arguably), and If everyone was listening are about someone who's on the edge of that horizon, and in danger of going over.
Many country and western songs, especially Johnny Cash's songs "Folsom Prison Blues," which is about a man sentenced to life imprisonment and "25 Minutes to Go," about a man who is about to be hanged.
"Scarsick" by Pain of Salvation follows a man who grows increasingly frustrated by the various facets of modern society shown to him through television. Eventually, he decides he's had enough and jumps off the roof of a building in an attempt to shock the people around him back to their senses... whether or not this works is left up to the listener.
Also, in the same vein, the extra track off "Entropia," Never Learn to Fly, a song where one of the characters decides that dreaming and striving for anything great will only lead to unbearable pain... at least Plains of Dawn had a hopeful point, however brief.
Roger Miller's "One Dying And A Burying": One man contemplates suicide to forget the pain of lost love.
"Dance with the Devil", by Immortal Technique. The protagonist rape a random woman in a dark street to be deemed "worthy" to integrate a gang, then is asked to shoot her as witness. It's his own mother... And they suddenly recognize each other.
Van Der Graaf Generator's "A Plague of Lighthouse Keepers" seems to be the self-narration of a man who cross the horizon, and then commits suicide.
The non-album single "W", where a man dies of extreme loneliness.
Radiohead seem to have built their whole career on this.
The Protomen: both Protoman and Dr Light cross this when the Robot Masters drag Proto Man down. Mega Man crosses it when he kills Protoman and discovers how pathetic and self-serving the people he was trying to save were. As a general rule, anyone heroic will either die or cross this particular threshold, maybe both.
"Waffle House" by David Wilcox portrays the titular Real Life restaurant as a haven designed to help its employees cope with whatever depressing event they're dealing with (be it heartbreak, or highway, or some altered state) and stop them from crossing the DEH.
When it's time that we slow up We wrap both our hands around our cup And stay until the feeling goes As long as there's broken hearts and dreams And all of this highway in between The Waffle House will never close
A few Iron Maiden tracks. Specially "For the Greater Good of God".
Their song "Hallowed Be Thy Name" is their most notorious example, telling the story of a prisoner in death row who is just hours away from his execution; he is at first calm and rationalizing about the upcoming ordeal, but as the song progresses he starts to slowly fall apart until he breaks and starts crying, realizing that he is afraid to die after all.
Tears fall but why am I crying?
After all I'm not afraid of dying.
Don't I believe that there never is an end?
The singer's character in Project Pitchfork's "Lament" has apparently crossed this line after being jilted (or his love interest dying, depending on how it is interpreted), and is preparing to off himself. "I lay down here, forever to sleep".
"Cemetery Gates" by Pantera is told from the perspective of a man who is going through a despair event horizon following the death of the woman he loved. For most of the song he is lamenting his loss, and in the final verse he is actually contemplating suicide so he can join her in the afterlife.
"Fade to Black" by Metallica is about a man who has lost the will to live. At the end of the song, he commits suicide.
"Square One" by Interface:
When the hope has gone away When the night has taken hold Emotions are no comfort No shelter from the rain Denied what has been offered An empty hand once more Gaining only to lose again This has all met its end Left here with more broken dreams No desire to begin again