Broken Base: The creation of the Super Junior M subgroup and the inclusion of Henry and Zhou Mi in said subgroup. One of the earliest splits in the fandom.
There's certain groups within the fandom called "13elieve" (supporting only 13 members of the group), those who support all 15... It's a very polarizing fandom.
Ensemble Darkhorse: Ryeowook's role as the spastic student council vice-president in the band's Affectionate Parody movie "Attack on the Pin-Up Boys" was a fan favorite and actually won him both praise from movie critics and nominations for Best Supporting Actor and Best Comedic Performance at the Korean Movie Awards.
Face of the Band: Originally Siwon and Kibum for album covers, now only Siwon. Kyuhyun is on his way to achieving this status as well.
For variety shows though, it's Eeteuk.
Funny Aneurysm Moment: During the Explorers of the Human Body program, the members pranked Eunhyuk and Ryeowook to make them cry tears of genuine sadness. They did this by pretending to start a fight over Shindong leaving the group to pursue a hosting career. After the fiasco had ended, Kangin said "we must never let this happen in the future," "this" being the breakup of the group. Fast-forward to 2011 - Kangin ended up taking early military service after his drunken assault, Han Geng sued their label and went back to China to start a solo career, and Kibum hasn't been seen anywhere since Sorry Sorry.
Germans Love David Hasselhoff: Henry and Zhou Mi are not popular in Korea, though this has died down a bit. However, they're two of the more popular members in North America.
Eeteuk. What with the sexist comments, his need to always be in the spotlight, and the infamous "blackface" incident, he's been getting a lot of rage. At the same time, he does try hard to be a good leader, and a combination the band splintering, the incredibly hectic work schedule, falling sales, and the impending two years' national service that might erase everything he's worked so hard for has really taken a toll on him. When news of his incredibly depressing tweets and videos of him bawling his eyes out during Heechul's last stage came out, fan reaction could be summarised as: Come here so I can give you a hug, you fucking asshat!
Portmanteau Couple Name: In a group of thirteen to fifteen men with a possibility of 105 pairings (not even going into the threesomes), this is inevitable. Examples include Kangteuk, Hanchul, and Eunhae.
Shipping: The members are well aware of this, particularly Eeteuk+Kangin and Eunhyuk+Donghae, and insist on playing it up.
Heechul has admitted to reading fanfics about him. Specifically, Slash Fic of him and Han Geng, who he apparently can't look at the same after...
Take That: The inclusion of M's Perfection on the fifth album as a bonus track (a song that heavily features Henry and Zhou Mi) looks to be a shot at the Only13 portion of their fan club.
Tear Jerker: How many times their pranks on each other have ended in tears?
The two car accidents, especially the one that almost killed Kyuhyun.
Kangin enlisting in the military.
Hankyung filing the lawsuit.
Also, the depression many of the members have experienced.
Donghae's dad passed away from cancer shortly after Donghae debuted as a singer and before Donghae could even give him his first paycheck. Donghae was unaware of how serious his father's condition was at the time. He says his greatest wish is for his father to see him perform on stage and to sing for him just once The members have shed tears over the passing of his father as well.
Although Shindong's even been called out by Korean fans for acting like a moron on an awards show (going behind winner IU and making stupid faces even after someone tried to pull him back - twice!), and commenting that women should be skinny and take care of themselves.
There's also values dissonance on the part of the fandom - Shindong was revealed to have had plastic surgery, and pretty much the entire SJ fandom cooed over it - a far cry from how they treat female idols who admit to it! (To be fair, female k-pop idols have far more rigid standards than male.)
The Woobie: A majority of them due to being overworked, underpaid, with few hours to sleep, crazyfans, and rarely getting to see their families. Eeteuk, the overworked and always depressed leader. Donghae, whose father died and has to put up with talking about it in more interviews and talk shows than necessary leading to the poor guy crying every time. Kyuhyun who almost died in a car accident. Henry and Zhou Mi who split the fanbase.