Romania and Spain of Axis Powers Hetalia remain optimistic, upbeat and cheerful, despite the fact that they've experienced many unfortunate and turbulent events in their histories.
Elmer C. Albatross from Baccano!. The woobie factor is subverted in that most people find his constantly chipper, love-everyone attitude to be inappropriate and incredibly insensitive: "C'mon, let's laugh together!" is not what a teenage girl wants to hear just after nearly being killed by her fiance's murderer. Pretty much the only person that doesn't find him creepy is the equally creepy Huey Laforet.
We eventually learn that his Pollyanna nature is actually a front. Not that he feels sad about anything. Just the opposite: he's a sociopath. He doesn't care about anyone, but he wants everyone to smile just because.
Bleach: Orihime Inoue is ultimately a subversion. She seems to be this in the first arcs, but she's later revealed to be a Stepford Smiler who hides her massive self-esteem and lonely issues behind her smile... and once she's taken out of her more-or-less safe Karakura environment, she's subjected to a Break the Cutie process. She arguably plays this straighter when she's back home, but it can also be seen as a coping mechanism: she does state that she's had a rough life, but is trying not to dwell on it anymore.
Laios of Delicious in Dungeon is perpetually cheery and relaxed, which can be jarring as he casually talks about how he cooks and eats demons from the underworld and plainly comments how his sister was resurrected using extremely dangerous dark magic. It's not that he doesn't worry, it's just that his near unshakable demeanor clashes horribly with the mood.
D.Gray-Man. It's very, very difficult to upset either Allen Walker or Lenalee Lee permanently. Allen in particular has been put through staggering amounts of crap by a universe that clearly has it in for him, but is nevertheless an All-Loving Hero and a Wide-Eyed Idealist who wants to save everyone.
Goku of Dragon Ball. He is an unrelentingly optimistic and happy person. At the start of the series, he tells Bulma that his parents abandon him in the woods where his grandfather found him. His grandfather died years ago, stomp to death by a giant monster, and he was left alone in the mountains with no human contact the entire time. He says all this with a huge grin on his face and never shown any visible signs of trauma. Bulma outright ask him how can he be so happy. Even during the most hopeless of times, he can be seen smiling, despite saying he's afraid.
Nana from Elfen Lied is clearlybroken by the time we meet her and continues to have horrifyingly terrible things happen to and around her for the entire story. She nonetheless somehow clings to a generally positive outlook that everything will be okay, especially as long as she can be with papa again. She doesn't. He explodes, though he survive in the manga, version.
Excel Excel, the ultimate oblivious Genki Girl, who will always believe that she is Il Palazzo's favorite minion no matter how violently he rebuffs her, until Il Palazzo shoots her at the end of episode 23. Although he has shot her before, including with antimateriel weapons, during this episode the Toon Physics are turned off.
Tohru Honda from Fruits Basket. In the anime, the other characters use her as a doormat, but she remains eternally perky and eventually wins them over. In the manga, on the other hand, while she acts the same, she's also shown as doing some serious repression.
Mikan Sakura from Alice Academy. She's eternally optimistic, and happy. She's the EXACT DEFINITION of the Pollyanna. Even after Mikan discovers her alices and all the horrible stuff that happens to her, she always cheers everyone up. Even Natsume and Luka love her, eventually. And Natsume Truly and deeply loves Mikan. But really, who CAN'T love Mikan?
Milfeulle from Galaxy Angel is almost impossible to make depressed or sad for very long, even when starving to death, being completely broke, having people actively trying to kill her, being dead...
Lacus Clyne, from Mobile Suit Gundam SEED is a borderline example. She manages to maintain her childish idealism in spite of everything that happens, including her father losing his office to a genocidal maniac and then being murdered by him, along with most of his supporters and nearly Lacus herself, the nuking of several colonies and just generally living through a Gundam show. She did cry when she told Kira about her dad and was very upset when her Body Double Meer took a fatal shot for her, but other than that, she rarely breaks down.
Rico from Gunslinger Girl fits this trope, being overly happy with her new life as a cyborg-assassin, although it can likely be attributed to her aggressive mental conditioning.
In Hayate the Combat Butler, we have the titular Hayate. He has gone through some seriously heartwrenching moments as a child including: being sold off by his parents for frivilous reasons, being ostracized in school for having such parents and people assuming he's the same way, leaving his first and best friend in a fight caused and not being able to apologize because they were seperated, and having to work various degrading odd jobs to support his parents' gambling habit. Perhaps because of this, he strives to be the best butler he can be and flat-out refuses to burden people with his problems, so he smiles; whenever someone asks him about his past, he always answers cheerfully and never complains.
Hayate has actually gotten mad at and expressed hatred for his parents a few times.
The eponymous character of Helen ESP, who doesn't let being rendered blind, deaf, and mute and her parents dying get her down. Though, this is after some backstory Character Development from being a very bitter girl.
InuYasha: Within the first chapter of her introduction, 7-8 year old Rin is revealed to be an orphan who watched her entire family be slaughtered by bandits in front of her eyes (a trauma that left her mute and continuing to suffer nightmares), who is scrounging a living on the outskirts of her home village, being beaten by the villagers for stealing fish to survive... and then she's murdered by wolf-demons. After she joins Sesshoumaru's group, she continues to be a target for kidnapping and attempted murder. Despite everything that's happened to her, she remains a constantly cheerful, upbeat person to the end.
Sora, protagonist of Kaleido Star, has a will of steel, considering the hoops she's had to jump through to get to the top (both figuratively and literally).
Sora is actually a massive subversion. Despite being super sunny and plucky, she does have several breakdowns ( to the point of temporarily leaving the Stage in the second season), and later has to settle for a somewhat less rose-tinted view of the world. Considering she's a massive deconstruction of Purity Sue, it makes sense.
Though he has currently (as of the Time Travel Arc) been told the truth, and actually knows more about certain backstory elements than any of the main characters, not to mention the readers.
No matter how many times Shibuya Yuuri from Kyo Kara Maoh! is betrayed, he is always willing to give that person a second chance. When a girl is jailed for posing as his daughter and trying to assassinate him, he smuggles her out and adopts her.
Nao Kanzaki from Liar Game. Altough when things go real bad, she panics to the point of tears.
Negi of Mahou Sensei Negima!starts as one of these, but when he begins training under Evangeline, she specifically makes it a point to knock that out of him, because the world is not that nice, and he needs to realize that. As it turns out, she wasn't entirely wrong.
Jean from Nadia: The Secret of Blue Water is a shoe in for a inventor's competition which he ditches to save Nadia, and not only does his plane crash but he is perpertually on the run and in constant danger. And yet, through it all, he is cheerful and even enthusiastic about the technology from both sides in general. (Only exceptions is when he and Nadia exploring Gargoyle's base, and even reacting in horror to two deathsone of an escaped fugitive, and another, a crew member. This latter incident causes for a Break the Cutie moment that turns into a Heroic BSoD when he learns his father is dead. Nadia cheers him up by reminding him of his promise to her and encouraging him to build a gyrocopter.)
Brook from One Piece is an interesting male example of this trope. One of the characters almost Lampshades this by asking why he's so cheerful even though his life has been awful. It's heavily implied that Brook has gone insane from all of his trouble, though.
Really, most of the Strawhat Pirates fit this. Luffy fit this partly before the timeskip, what with witnessing Sabo dying, while trying to become a pirate, and still not giving up on his dream of being the Pirate King. After the timeskip he fits this even more, since he saw his brother, Ace, getting killed right in front of him, and he still managed to push past it and have a positive outlook on life and keeps following his dream..
Ash from Pokémon, after the Hoenn season was over, or specifically after his feud with Brawly. The guy has hardly lost his sunny disposition despite suffering many times in his adventure, though this is extremely exaggerated whenever he loses the chance of winning a league and he just keeps smiling and never considers having a rematch outside the league (contrasting heavily to how he was an emotional wreck when he lost the Kanto League). That is, unless he constantly butts heads with a Jerkass rival that riles him up for being a disrespectful prick to the title creatures or him.
Pikachu himself is another prime example.
Just about every creature is like this, unless they like being a lone wolf or they have a much more serious attitude as a late-stage evolution. It has been proven many times that being "adults" isn't enough to exclude them from the attitude and still may act like morons.
On that note, unless you're not the Jerkass rival, you are this by default, invoking a Sugar Bowl.
Kasumi Tendo from Ranma ½. Nothing catastrophic happens to her, but she does seem to find herself witnessing catastrophe on a regular basis. Unless you count being possessed by a demon as catastrophic...
Nodoka Saotome is farmoreawareofher situation, but refuses to let it get to her, making her optimism seem heroic next to the passivity of Kasumi (who lives in a big house, surrounded by family, not a care in the world.) At least, until she starts crying in her sleep... and reenacting a Seppuku ritual.
Usagi of Sailor Moon isn't as much of a Pollyanna as fanfic writers would have you believe (where Usagi is a perky saint and can never be pried from her hold on a "good in everyone" mentality), but she does have an overwhelming belief in the power of love in the anime. The manga and live-action are bit darker, though.
Takeru from Snow White and Seven Dwarfs. While he does have his moments when he's upset, he's generally upbeat and cheerful despite being in a Crapsack World, making it a point to try and remain positive no matter what happens.
Subverted in Sonic X by Cosmo, whose entire family was wiped out by the Metarex right in front of her and who has frequent feelings of self doubt, Survivor's Guilt, shame for her own hatred of the enemy and unworthiness.
Komachi Shokichi from Terra Formars, in spite of having lost his childhood sweetheart to a Mars expedition gone wrong and witnessing his entire team die, does not let them beat down his laid-back, friendly nature toward his subordinates and his determination to make things right for humankind in Annex I expedition.
Keiichi Segawa from the X1999 manga. Lost his father in early childhood, and then his mother is killed in one of the earthquakes caused by the Dragons of Earth, yet he remains optimistic and kind until finally being phased out of the series.
Butterfly from Mademoiselle Butterfly. Sold as a child to a geisha house, facing the perspective of getting married to a stranger and never seeing again his childhood love, getting thrown into jail as a way of blackmailing said love, and after getting married having to get separated once he gets drafted into war aren't enough to break her spirit.