Momo does this out of insecurity sometimes, and later on after she thinks that she was raped (she wasn't).
One scene had Sae hiding from a porn star and his camera crew who wanted her to film with them. She thought it was a video game shoot but Ryo lied to her about it. She hides in the bathroom, crawls into the bathtub, and starts sobbing while cuddling her legs close to herself. Thankfully Momo and friends save her before anything happens.
SHUFFLE!: After Kaede snaps, she is shown taking a shower in this position.
Nanami goes into the Troubled Fetal Position after discovering that Akio and Anthy, who are brother and sister, have been regularly having sex.
Utena is seen in this pose in episode 10 while lamenting that perhaps she, being a girl, can't become a prince after all. She does it again in episode 12 after losing the duel and Anthy to Touga; in the same episode, Wakaba does it after her fight with Utena.
And finally, there's a flashback where the younger Saionji and Touga find a little Utena like this inside a coffin, totally traumatized after her parents' deaths. This scene inspires another where the older and no longer kind Saionji sees Anthy curled up in a coffin full of roses, mimicking what Utena did...
The pose is exhibited by a very young Misato (The Birth of Nerv), few after being the Sole Survivor of her father's crew. In fact, the just as young Ritsuko saw her for the first time when she was like that.
Before that, Naru is seen twice like this. The first time she's sitting on the bottom of a slide on such a position, as she and Usagi discuss Sanjoin/Nephrite's actions against Naru. The second, she's cutrling up in her bedroom after Nephrite's Redemption Equals Death as Usagi and Umino try to comfort her.
At second witches' tea party, Bern and Lambda make a few different comments ("What? Are you still hugging your knees?") that imply that Battler wound up doing this after all the unspeakable horrible things that happened in the second arc.
At the beginning of EP4, Ronove alerts Beatrice that Battler has assumed the position once again after her fake Heel Face Turn at the end of EP3 ("he has been crouching and clutching at his knees for some time now. I spoke to him several times, but he did not answer. I also carried him food, but he made no effort to touch it"), and that he probably won't be able to continue the game unless she snaps him out of it. Battler's just sleeping in the fetal position, but why tell Beatrice that when you can trick her into making an idiot of herself trying to cheer him up?
While Soul Eater's Crona has sat in a Corner of Woe (played for humor), there are instances when he/she has done it in a more serious sense. One involves a scene where we see Crona inside of his/her soul (or mind) which is shown as an empty desert. Crona sits in the fetal position and his/her shadow asks all sorts of questions in which Crona answers with "Pass." At one point he/she sobs into his/her knees. Considering Crona's past with his/her abusive mother Medusa, it's no wonder.
Another notable scene features 2 Meisters and their Weapons in this position after reading Crona's poem and (s)he soon after follows suit.
Soul Eater has Professor Stein a scene where he's curled up in the fetal position, begging for someone to make the noise in his head go away.
Renton & Eureka does this in Eureka Seven whenever they are troubled or sad.
Aine does this in chapter 8 of Sensual Phrase when worried about whether or not she can remain as Lucifer's Lyricist.
In episode 11 of Ghost Stories Satsuki ends up doing this after spending just about the whole episode being stalked by an extremely Creepy Doll.
In Gankutsuou Albert does this while taking a Shower of Angst. He later does this after his engagement to Eugenie had been canceled and she becomes engaged to Andrea against her will.
The Sgt. Frog anime has Dororo often take this pose while in his Corner of Woe. Normally Played for Laughs. The episode that introduces the Soul Diver plays it a bit more seriously, as he gets stuck in his Trauma Mode and has to be pulled out of it, with Dororo staying in this position the whole time.
At the beginning of the Skypiea saga, Usopp nearly drowns on the way up through the Sea Clouds. Then he jumps out thinking he's going to swim, only to fall right through, barely saved by Luffy and Robin. He then spends half of Chapter 237/Episode 153 curled up in a fetal position muttering "sky island so scary".
In a serious variant, there's King Neptune, who went into this after his wife, Queen Otohime, was assassinated, as he desperately attempts to hold back his murderous rage over it so as to avoid ruining everything she spent her life working toward.
Trigun: Vash shoots Knives and then runs away after Knives first tests his angel arm. Knives clutches his wounded thigh and cries, "My leg! Why is there so much pain in my leg? Did you shoot me? DID YOU JUST SHOOT ME?!" while he and Vash are looking at each other in horror. Then Vash lets out an almighty scream and runs away. After that, Knives curls up in a fetal position while crying his eyes out.
Mio Hio in episode 20 as she grows more troubled about having to sacrifice Daisuke for a chance at truly being human, all while the time in her current borrowed body is running out.
Attack on Titan: Extreme Doormat Bertolt seems to do this as his default when sitting down. Also a common occurrence whenever something horrible happens, with both named characters and random extras assuming this position as despair or trauma take over.
In Kimi ni Todoke, Ayane is briefly shown sitting like this after breaking up with Mogi.
In Digimon Tamers, Jeri Katou spends a lot of time curled up like this while she's held prisoner by the D-Reaper.
Aya of Tantei Team KZ Jiken Note does this in the part of The Valentine Knows animated in episode 15, after finding her Valentine card to her crush Sunahara being posted on the train station's notice board, and overhearing that it was Sunahara who posted it.
In the Another anime series, it happens at least thrice:
In the flashback to how the Calamity took place in 1983, when a boy named Jun Hamaguchi is fatally hit by lightning, the girl who was walking right in front of him does this to shield herself. She might or might have not been Yuki Hoshikawa, the victim who followed.
Towards the end, Matsuko Arita"assumes" the pose◊ after she barely survives the incredibly messy slaughter of her classmates.
Whenever Kyo is genuinely upset, he runs off to be alone and does a variation of this.
Rin is seen in this position more than once, including a manga cover.
Koshiro from Koi Kaze is seen like this after learning that Nanoka, who he has essentially just had a date with, is actually his younger sister.
Chihiro does this on the stairs, while saying "Go away, Go away... Disappear..." after the events of the beginning of Spirited Away have unfolded.
The last scene before Full Metal Panic!'s epilogue is of Sousuke curled up and sobbing in the blown-out hole that used to be an AS's cockpit, realizing that he was seemingly going to die alone without even getting to say goodbye to everything he had finally recognized was Worth Living For.