There was definitely was a
catalyst that sent each of them into Despair. I've got my own thoughts on how it went down. (Note: The following is 99% conjecture)
He became Kamukura. With no hopes, no dreams, no aspirations, no happiness, it was natural that someone like him would fall into Despair. Being forced into the killing game that kickstarted the Tragedy probably didn't help either.
Junko. Enoshima. Nagito is someone who values hope to an extreme degree. He's always believed that Despair is merely a stepping stone for Hope; the greater the Despair, the greater the Hope, and Junko is Despair incarnate. He most likely wanted to prove to her that Hope would overcome any Despair. Unfortunately for him, someone like him could never Hope to overcome the Ultimate Despair.
Mikan has spent most of her life dealing with abuse and mistreatment. Up until she met Junko, she hadn't met a person who truly valued her. Even her "so-called" friends at Hopes Peak didn't treat her all that well. Junko on the other hand, was someone who accepted and loved her. (Naturally, Junko used those feelings as a means to convert her into Despair). She fell madly in love with her and devoted her entire life to serving her. No brainer.
Fuyuhiko lost his little sister at Hopes Peak Academy. I don't care how tough Fuyuhiko props himself out to be; losing his sister must have sent him into unimaginable Despair. Junko probably used that anguish and anger to convert him into the Ultimate Despair.
As Fuyuhiko's tool, she was obligated to protect and follow him no matter. If Fuyuhiko were to join the Ultimate Despair, then Peko would follow him without question.
Just like Fuyuhiko lost his sister, Mahiru lost a friend. Someone she obviously cared about dearly, as she was willing to go as far to cover up her crime and immediately cried when she found out she died. Burdened with the guilt of covering up a crime and losing a close friend, she probably fell into Despair quite easily.
He's got intense identity issues, which can definitely lead into Despair, but my guess (and this is a longshot) is that since he was admitted into Hopes Peak Academy before
Byakuya as the Ultimate Heir, he probably fell into Despair after the real Byakuya was admitted into Hopes Peak.
Teruteru deeply, deeply, deeply cares about his mother, to the point where he'd murder someone just to see if she was still alive. Since his mother was in poor health prior to going to Hopes Peak, her death could have easily triggered a breakdown into Despair. And knowing Junko, she wouldn't hesitate to capitalize on that Despair.
If you do her FTE, you find out that Akane values her own survival over anything else. She lived in a poor environment were dying wasn't exactly uncommon, and she worked to support her various half siblings. But her main motivation came from the fact
that she just didn't want to die. She didn't want to be a weak individual that would die without a fight. I could see her choosing to side with the immensely powerful Ultimate Despair if it meant living, rather than die like the majority of students at Hopes Peak academy.
This one's also a longshot, but she probably fell into Despair following the breakup of her band. Music is her livelyhood, and if she couldn't write songs or perform, I could see her falling into Despair over that. But then again, it's
Ibuki. It would take a huge amount of misfortune for to join up with the Ultimate Despair.
Meeting Junko Enoshima probably triggered it. If you do his FTE, you find out that Soda's the type of person who did favors in an attempt to get people to like him. He let a kid cheat off of him in a test, for instance, and wasn't miffed when that very same student screwed him over. I could see Junko asking him to build the various execution devices, and in return she gave him what he craved the most (not sex mind you, you find out that he rejected women that threw himself at him) but someone he could call a "friend."
...That's how I see things. Can't really come up with a good enough reasons for the rest.
