Aang early on in Avatar: The Last Airbender. In the first episode, when presented with the knowledge that he has been frozen for a hundred years, a world war had broken out during his absence, and everyone he ever knew or loved is dead, he comes to the conclusion that meeting his crush more than makes up for it. His blissfully carefree nature gradually wears down over time, culminating in mid-second season where Appa is kidnapped, he has a Freak Out and spends the next couple of episodes trying to find ways to cope. One way is him hostile and overly violent, the next episode has him trying to be completely emotionless.
Uncle Iroh also counts. His son was killed in a battle he lost, his reputation was destroyed, he lost his rightful place on the throne and was banished from his home, his beloved nephew treats him like dirt, and yet he not only remains patient and cheerful but is eventually able to help his nephew solve many of his personal problems without even breaking a sweat.
It's implied that the first four experiences had a huge impact on him becoming the wise, collected uncle he is when the show starts. That doesn't change the fact his cheerful demeanor helps numerous characters, his nephew especially, over the course of the show.
The titular character of Eek! The Cat never loses his optimistic outlook or his Catchphrase "It never hurts to help". Even though every time he utters it, it inevitably hurts. A lot.
On Goof Troop Goofy ends up physically injured, screwed over by his neighbor, or unappreciated by his son on a regular basis, and has a disposition so sunny it can really only be compared with that of the remarkably fortunate Pistol. However, he does occasionally have breaking points, especially in the movies.
Lila from Hey Arnold!. Don't agree? Go watch the episode where we learn about her home life... just... watch... However, Word of God says that she is a Stepford Smiler, so it's likely she's just trying to seem optimistic and happy.
Eugene in nearly all situations. He's the show's jinx. His usual action after an insane run of bad luck-based injuries? "I'm okay."
Phineas and Ferb has the first of the titular characters exemplifying this trope most of the time. Phineas is extremely hard to make angry. In fact, the only episodes in which he tends to be angry are special two-parter ones.
Lampshaded in one episode
Buford: This must be a special episode. He's yelling at his sister again.
Ms. Frizzle, the teacher from The Magic School Bus. Even when the bus and her class are getting baked into bread, assaulted by predatory animals, or attacked by a robot while on their crazy field trips, she more often than not has a smile and cheerful demeanor as she deploys a quick, magical way to get them out of the situation.
World of Winx has Cliff, one of the two judges of WOW, a reality talent show, is cheerful and supportive of the contestants . . . in marked contrast with fellow judge Margot, who is, well, a Grumpy Bear.
The Oblongs — Living in denial, played for all the black humor the writers can get out of it. Bob Oblong, the limb-less husband and father is the straightest example of the trope.
Greg from Over the Garden Wall. No matter what dangers they face, he's always cheerful (or, occasionally, snarky). Even when Wirt convinces him that their troubles are all his fault, Greg doesn't dwell on it—he simply gets to work trying to fix everything. The Beast even seems to realize that he can't push Greg toward the Despair Event Horizon like he can with Wirt, and tries to freeze him to death instead.
Butters from South Park. This is actually mildly subverted in his "Very Own" Episode, which depicts his parent's anniversary and their annual trip to the restaurant Bennigan's. His mother asks him to tail the father to see what he is getting her, Butters finds out his dad is gay without realizing it himself (he describes what he saw and his mother puts it together), driving his mother mad, and prompting her to kill him and commit suicide. Both fail, but she doesn't know that the former did. The rest of the episode is the parents pretending that a stranger kidnapped their child, and when Butters shows up and everything is admitted, he is fine, saying "When I have a chipotle blu cheese bacon burger at Bennigan's, I'll forget all about my dad being queer and my mom trying to kill me." "Really?", he is asked by the other kids. "No, I'm lying." (His happy face barely cracks, though.)
Spongebob Squarepants can also come off as this. He tends to be very positive and optimistic about most things such as the beginning of summer, his job at the Krusty Krab (he even begged Squidward not to quit in one episode where Mr. Krabs accused the latter of stealing his dime, because "What could be better than serving up smilies?", Squidward replying "Being DEAD or anything else!") Most things in life don't bring down his mood unless it's extreme such as when Nematodes eat up his house or he failed his boating exam 38 times.
Ilana in Sym-Bionic Titan. Her home planet's in danger, but she still tries to make the best of living as an Earthling, even when her classmates aren't very nice to her.
The character, Daria, is like this in The Princess and the Pea. She remains cheerful in spite of doing all the chores on the farm as well as taking care of her lazy, unloving step-parents. This is very evident when she prays for what she's thankful for before going to sleep in the chicken coop she sleeps in every night.
Lindsay is usually one of the more optimistic campers, usually because of her stupidity.
Izzy, because of her craziness, is a borderline parody. She takes being strangled by a snake in stride.
Zoey is a downplayed example; while she's one of the more optimistic and idealistic castmates within the second generation cast, she does have a breaking point.
Despite numerous attempts to break her, Ella usually maintains her sunny disposition. She even takes her elimination in stride, singing a song before she is shot out of the Cannon of Shame. However, as a Double Subversion, she is visibly depressed when she realizes that Dave is attracted to Sky and not her, but she gets better, and still remains friends with both Dave and Sky.
In the Animated Adaptation of Where's Waldo?, Waldo would gleefully and knowingly walk into perilous adventure with the cheerful demeanor he always wears in the books.
Almost to the point of Obfuscating Stupidity at times. There was one instance where he was confronted with a vampire, and, without changing his expression or even blinking in surprise, simply withdrew from his pack a jar of his family's special pasta with extra garlic, gulped it down in one gulp, deliberately gave the vampire a huge dose of his now horrendously garlicky breath (Hhhhhhhhhey, there. Hhhhhhhhhhave a nice daaaaaaaaay?), then just happily went on his merry way as the vampire fell over, defeated.
Disney's Goofy as a character is ready to acknowledge that he is a klutz, he knows he's not the sharpest card in the deck, and even knows that people sometimes take advantage of him, but he simply doesn't care what other people think. His most consistent goal in life is nothing more than to love and be loved, and he becomes one of Walt Disney's most beloved characters as a direct result.
Kaeloo: Kaeloo always tries to be happy and see the positive side of things (although this sometimes leads to her being unaware of the chaos around her). For example, when she gets Trapped in Another World with bizarro versions of the people she knows, instead of panicking about not being able to go home, Kaeloo celebrates the chance to spend time with her "new friends".
Perfuma starts out like this, insisting that she's sure the Horde poisoning her lands is going to be fine because the universe will restore balance sooner or later. After Adora fails to solve the problem with She-Ra's power and instead takes a more direct approach, Perfuma realises that hoping it'll all work out isn't going to help, and leads her people into battle, vowing to rebalance this part of the universe herself.
Scorpia is a more tragic version. When she relays her backstory to Emily, it's a shaky structure laden with tissue-thin rationalisations: it's for the best that her people gave away everything to the Horde of their own free will and got back she doesn't know, something, she guesses; her grandfather must have known, when she was an infant, that she'd be happier as a soldier; the Horde are the only people who would accept her anyway; and Catra is the best friend she could possibly have and will no doubt recognise Scorpia's merits and start being a better person any day now. When Emily directly challenges the last one with a hologram of her genuine friendship with Entrapta, the whole thing comes crashing down, and Scorpia becomes a wiser person...and, after finding friendship and acceptance with the Rebellion after her HeelFace Turn, a genuinely happier one too.