The first chapter of Ana Satsujin shows our main character, Kurosu, try to commit suicide by hanging himself on a wall-mounted hanger. However, the hanger tears out of the wall due to shoddy construction, creating a peephole into his neighbor's bedroom and acts as the catalyst to the story.
Berserk provides a particularly depressing example with Griffith. After enduring a year of Cold-Blooded Torture that left him permanently crippled, seeing all his dreams and ambitions go up in smoke, and finding out that both his Love Interests hooked up with each other, he tries to kill himself, only to find that he can't even do that right. Unfortunately for the entire cast, this sends him over the mother of all Moral Event Horizons.
On Blue Exorcist, Yukio's self esteem and ressentiment issues get so Bad that They culminate in this. He is saved by a mysterious phenomenon which gets to rare exemple of this trope being invoked: He attempts to commit suicide again to find out about what saved his life (intending to die for good if that fails).
Jeremy tries to throw himself into a lake after finding out his mother knew he was being raped by his stepfather. Good thing Ian has ears like a fox, heard it, and managed to find him through the sleeting rain and fish him out. Jeremy also has two failed suicide attempts after returning to Boston while he is living with his aunt.
Sandra has two of these, one pre-series and then another after Greg cancels their wedding.
Marjorie has quite a few failed suicide attempts. The most notable of these is when Pascal locks her in the bathroom while he beats Jeremy in a drug induced rage. She tries to drown herself in the bathtub.
After Leomon's death, Jeri falls into such a deep depression that she basically goes to the D-Reaper hoping it will kill her. Unfortunately,itdoesn't...and then things get alotworse for everyone on the show.
Ogiue in Genshiken joins the group after hurling herself out a window (and breaking her arm) following an altercation with the Manga Club. Later it's revealed she jumped off the roof of her old school after her "friends" ruined her life. It's not stated how she survived or what injuries she incurred.
Punpun from Goodnight Punpun shoots himself in the eye at the end however survives. He loses the eye.
The second chapter of Gunjo has a woman who tries to kill herself after the death of her son and fails. She succeeds after another try.
In Half & Half, Shinichi and Yuuki want to commit double-suicide before their 7 Days are up and jump off a cliff together. But the same light that saved them at the beginning keeps them from dying, so that they have to wait the remaining days out.
In Hoshi wa Utau, this is what happened to Chihiro's previous love Sakura. Her attempt left her comatose in the hospital.
Kaze Hikaru has two: Okita's first love and the 10th unit's captain Sanosuke Harada's historical attempted seppuku. The latter is Played for Laughs, as it was years before the story starts and Sanosuke only did to prove he could. He brags about the scar (as he did in Real Life).
One of the bullied characters in Life attempts to kill herself by jumping from a school balcony, but only ends up fracturing her foot and hurting herself.
In the very beginning of the manhwaMy Lovable Fatty the protagonist tries to hang herself but ends up breaking the rope.
It's implied that Asuka cut her wrists in episode 24, and is seen in a bathtub with water that looks like it's stained red. It's also implied that she was starving herself. She survived only because Section 2 found her before she bled out and she was too weak to resist being taken into custody.
The first scene of End of Evangelion heavily implies that Shinji tried to drown himself but failed. After watching the rest of the movie, everyone will agree that it would've been better for everyone if he succeeded.
Kaworu's character design originally included scars on his throat and wrists to hint at multiple failed suicide attempts, explaining why he asks Shinji to kill him.
Parodied by Kumadori's inability to commit Seppuku (every times he use the ability "Tekkai" and save himself).
Played less for laughs and more for awesome by Kaido. The man has survived so many execution and suicide attempts that he regularly takes up suicide as a hobby, to find out if he can be killed. His Establishing Character Moment shows him jumping off a freaking sky island and coming out without a scratch despite leaving a huge Impact Silhouette.
Paranoia Agent has an episode dealing with 3 people who had made a suicide pact over the internet. Over the course of the episode, all of their attempts to kill themselves (jumping in the path of a subway train, hanging) are thwarted, and it's all Played for Laughs. Subverted in that at the end it's revealed that they were Dead All Along.
In one episode of Q.E.D., a graduate of MIT tries to mark himself as the math major with highest honors of his graduating year by killing three highest-honor holders in other majors, then attempting suicide, but fails.
Played for drama in Rurouni Kenshin - It's evident in the scars on Megumi's wrists that she'd attempted suicide multiple times before meeting (and joining) the Kenshin-gumi.