When Sunny tried to fight back against Taeyeon's attempted kidnapper. (A little perspective: Sunny's about 5'1 and 43kg, but she still gave it her all. Damn.)
Some might also say their song Star Star Star is this and an example of Crowning Music of Awesome, as it gives all the members (except Yoona) a moment to shine in a gorgeous ballad with almost no auto-tune. It's become almost irrefutable proof that these girls can SING. See it live.
This video of them doing "Genie" with no backing music. Granted, some members are a bit wobbly and off key, but they completely own the song!
Jessica recently passed out from exhaustion during one the performances of her musical. Did she lie down and rest? Nope, she came out and apologized (she had to be carried out, but she refused to let everyone down).
YMMV on this, but considering how much flack Hyoyeon gets for being the alleged "ugly member" of the group, both fans and anti-fans agree that Hyoyeon can (and probably has) dance circles around the other girls and is one of the best dancers in SM Entertainment. Hell, she was even placed in a unit made up of the ultimate best dancers in SM Entertainment (Other members included Super Junior's Eunhyuk, Henry Lau, Shinee's Taemin, EXO-K's Kai, and EXO-M's Luhan).
Her participation to "Hit the Stage" let her perform masterfully various dance styles, culminating with a win in episode 9.
Winning their first Music Bank award for a performance of "Kissing You" that was done after the group was involved in a car accident that forced them to run to Music Bank on foot to make it to their performance in time.
DreamConcert 2013. There's a reason why KBS had to edit the performances, the whole stadium was just that loud.
Seohyun stood her ground when SM wanted to fill her first solo mini-album with songs written by other artists. They remained adamant on the title track, but every other song is penned by her.
Taeyeon spent up to six hours on each song while she was recording "My Voice", her first full album. She had to be given an IV drip because she was pushing herself too much, but she kept going nonetheless.
From that very same album, the "No thanks baby" part in "I'm OK". What makes it awesome is that, during her solo concert, Taeyeon changed it to either "Bullshit baby" or "Fuck you baby" complete with middle finger raise.