The "Chicken Boo" sketches on Animaniacs had only one person realizing that Chicken Boo was actually a giant chicken in a ludicrously flimsy disguise. Nobody would believe them (or misinterpreted the use of 'chicken' as an insult, as in coward), and even when Boo's deception was eventually revealed (by some event unrelated to their efforts), they never got any credit for their insight.
Animaniacs also had Dr. Scratchensniff, who's powerless to either stop or tolerate the zaniness personified that is the Warner siblings.
Even more so Suki, when she is with the group. She's the only of the main and recurring characters who could be considered to be normal. She takes it quite gracefully, though.
Also Katara sometimes, most notably in the episode The Runaway, in which she is the only one who realizes that excessive scamming of scammers will come back to bite them eventually. Toph started the whole thing, and Sokka being involved is not horribly surprising, but Aang should have known better.
In Sequel SeriesThe Legend of Korra, Tenzin is this to the council as a whole. They don't seem to realize their decisions and actions are further proving the point the Big Bad is making, and he is even outright ignored. To further it, he also plays the role to Korra and (in a comedic sense) to his family.
Mako (and to an extent President Raiko) is becoming this in the second season. Midway through, he's the only one interested in figuring out who's behind a string of bombings in Republic City beyond making the presumption that it's all Northern Water Tribe, as everyone else is acting according to Varrick's manipulations. Korra's hell bent on getting revenge against the North for both Unalaq's machinations and the above misattributed offenses, Bolin's become wrapped up in new celebrity status, Asami's struggling to keep her company afloat and losing hope because of it, and even Lin Beifong is pulling rank on Mako in favor of two mustache-twirling detectives who don't seem to be doing their jobs. And of course Varrick is pulling everyone's strings in the name of War for Fun and Profit.
Between Eska being an abusive psycho and Unalaq shaping up to be some sort of Well-Intentioned Extremist regarding the Water Tribe's spiritual issues, Desna's developing habit of asking questions is putting him on the road to this trope within his family.
Captain Star's First Officer Scarlett is the one who actually gets the work done between an egotistical captain, a fish obsessed cook and an engineer whose isn't that crazy, in fact, he's perfectly sane. He has nine sane heads on a single body.
In CatDog, Mervis and Dunglap are the only normal characters in the series.
Shnitzel from Chowder is usually a good example, though he does a have a few lapses into the weird goings-on in the show.
Number 1: Abby, we both know you're the only one on this team with any real common sense. When I'm screaming that we're being invaded by tiny, kid-eating leeches falling from the sky, you're the one telling me it's only snowing.
Elise is probably a better example, given Chris tends to go along with Dan's schemes. Though she does have her moments of Not So Above It All.
Danger Mouse. He has to deal with a Cowardly Sidekick (Penfold) and an addled, near-senile superior (Colonel K). DM will either take the initiative and stop their banter or just throw up his arms in defeat.
Lee, the protagonist of Detentionaire, is this on his show. When he's not available, Tina tends to step into the role.
Although only sane person is stretching it, Foxxy Love on Drawn Together was once accurately described as "The only one here who's not completely retarded." Spanky Ham also fits this role after getting over his Early Installment Weirdness; he's often one to give a speech with the reasonable solution to an issue, like saying that old cartoons with racist stereotypes may be offensive, but denying them as part of our culture is dangerous.
On The Fairly OddParents!, Wanda is usually the only one aware of how insane everything is, or knowing of how bad everything is going to end up. This fact is often pointed out by Timmy and Cosmo, but they still continue with their antics despite that.
During the first season, and whenever fairy culture, rituals and quirks are involved, Timmy often assumes the role of the only sane man. As an outsider, he seems to be the only one to notice how weird that world really is.
Family Guy: Brian Griffin is often portrayed as the Only Sane Man, although everyone who isn't Peter ends up playing the Only Sane Man once or twice.
Hermes often takes this role most notably in Bender's Big Score where he's the only person to realize the crew is being scammed. He's also the most consistent critic of the incompetent Zoidberg.
Hermes' role is well exemplified in "The Duh-Vinci Code":
Professor Farnsworth: We must fly to Rome and exhume the body of Saint James! To the ship!
Hermes: Didn't we used to be a delivery company?
Professor Farnsworth: TO THE SHIP!
Leela is easily the most level-headed person, compared to Fry, Bender and the others. Though she has her moments, particularly when she is insulted. She holds very strong grudges.
Goof Troop has Peg and PJ both in this role; however, usually not at the same time, as Peg tends to have little screentime in PJ and Max episodes and PJ acts as a Satellite Character in most Pete and Peg episodes. Overall, Peg is more likely to be silly than PJ is, and PJ is more likely to misunderstand a situation than Peg is—despite this, both are usually the voice of reason for the others. Also, the characters are far more likely to listen to Peg than they are to PJ because Peg is far more assertive than her son is. Considering the rest of the cast consists of a Cloud Cuckoolander, an irrational Jerkass, a Fearless Fool, and a Genki Girl...
Lin Chung of Hero 108 is always calmer than the rest of the cast.
In Invader Zim, Dib and Gaz are both in this role, as they only ones who know that Zim is an alien, no matter how blatant it becomes. Gaz arguably even moreso, as while Dib obsesses over Zim's constant plots, Gaz is all too aware that Zim will screw himself over with or without Dib's intervention.
Originally, both of the Tallests were shown to be somewhat moronic, but Red's Divergent Character Evolution was sending him in this direction, best evidenced in "Backseat Drivers from Beyond the Stars".
Dukey from Johnny Test. He's the only one aware of the bad side effects of Susan and Mary's experiments (besides Hugh) and he knows when a fun time can get dangerous.
Kim Possible has Shego. Any episode featuring Doctor Drakken will have her standing off to the side making running commentary on the absurdity of his plan, or how it will inevitably fail because of one major overlooked factor. It is also implied in the episode 'Go Team Go' that this was the reason for her leaving the team of superhero siblings and going bad.
Hank Hill of King of the Hill is a shining example. In his version of the Rashomon, he sees himself as the only adult among children, and in a hallucination caused by varnish fumes, he imagines the rest of the world as helpless rodents on a Whack-a-Mole board.
Boomhauer also plays this role too. While he certainly does pick up the Idiot Ball on several occasions, he's easily the most rational of Hank's friends. Heck, Hank barely has to babysit him at all, he even complained once about not being in hot water with Boomhauer since it would be less taxing than Bill and Dale.
Bugs Bunny, of all people, spends time in this role in episodes of The Looney Tunes Show. Mind you, when you spend most of your time with a) Daffy and b) Lola, it's hard to avoid this characterisation.
Charles Offdensen from Metalocalypse. We actually see him go through the three stages in the background; by the time season two ends, he's learned to roll with it. Season three sees him realize that he's not the Only Sane Man. He's just the Only Sane Man on his side of the conflict.
Tommy Anybody from Mr. Bogus invariably falls into this role, when having to deal with the wacky antics of Bogus (and occasionally Brattus).
Of the Mane Cast, Twilight Sparkle usually plays this role, often being bewildered by her friends' antics, and patiently offering impartial observations and sound advice (that is only sometimes followed). She is, however, not immune to occasionally being tossed the Idiot Ball or suffering Sanity Slippage; at which point, Spike will take the role. That said, circumstances occasionally allow any the Mane Six (yes, even Pinkie Pie) to hold the Only Sane Mare position.
Twilight Sparkle: All the ponies in this town are crazy!
When Twilight or Spike aren't this, it's usually left to Applejack to fill the role.
Of the Cutie Mark Crusaders, Sweetie Belle, in spite of being the slowest to catch on, is normally the one to criticize the more dangerous and ridiculous plans the group comes up with. She's the only one who suggests "Tell an adult." as a course of action when a problem gets too difficult for the CMC to solve.
Of the pets, Owlowiscious is easily the smartest and sanest. In "Just for Sidekicks" he's the one who helps Spike keep the other pets in line, and in "Inspiration Manifestation" he tries, in vain, to warn him against using the Tome of Eldritch Lore.
Ned's Newt: Ned often acts like this, being bewildered at how crazy the people around him seem sometimes.
And occasionally Private. For example: "The Red Squirrel." And Marlene, all the time (except when she's outside the zoo grounds or talking about Spanish guitars). Rico once, actually. In "The Hidden" he keeps Kowalski and Skipper focused on the task at hand (finding their friends). Skipper, in any episode where the penguins encounter anything that terrifies them like in Snakehaed! or in any situation where he dope slaps all three of them at once. Kowalski, almost never. Though he thinks he is. Julien, no way. Mort, uhh... No...
Karl Pilkington of The Ricky Gervais Show. While Ricky and Steven laugh about all his thoughts, a fair number of his questions and statements are perfectly logical: notably a thought experiment where he wondered if one would be able to confidently state that they are the original if they had a clone with all the same exact memories up to the point of their creation.
Rocko from Rocko's Modern Life seemed to start out this way, a hapless bystander trying to use logic to navigate a thoroughly insane environment and oftentimes failing. Strangely, he seemed to lose this as the series also took a turn away from being as surreal as it once was.
In Sealab 2021, Doctor Quentin Q. Quinn is the Only Sane Man. At least in the beginning of each episode. Often the sheer insanity of everyone else becomes contagious or he would get fed up with the insanity and blow up Sealab.
The episode "Homer's Enemy" featured the character Frank Grimes, who is the only one of Homer's co-workers who recognizes his pathological boobery. He tries to point it out to everyone, but to no avail. He eventually works himself up over it so much that he has a nervous breakdown, during which he accidentally electrocutes himself. After some fans complained his treatment seemed a little mean-spirited, DVD commentary explained Frank was mainly a device to show a normal person couldn't function in the show's universe.
Lisa Simpson also often finds herself thrust into the role of only sane...girl. Quite often, when her family (or occasionally, the entire town) is caught up in some insane course of action, she's the one who points out how absurd things have become. Her comments are usually ignored. The straight man role is usually traded between Lisa and Marge. Sometimes Lisa is also caught up in the craziness, for example with the Movementarians.
There are even times that only the Simpson family really know what's wrong, and have the guts to do something about it, putting the whole family in this role.
Lou is the only sane man in the police force in Springfield, and frequently points out Chief Wiggum's incompetence or stupidity.
Carl Carlson is this, at least in comparison with his friends.
Other characters to have this role on a semi-regular basis include the more conventionally competent characters such as Apu (and his brother Sanjay is this to him when he has gone a bit too far with his work), Dr. Hibbert, Superintendent Chalmers, Sarah Wiggum (within the Wiggum family - "Clancy, use the remote") and Waylon Smithers.
Word of God states that Chalmers and Grimes were written as the only people who could function outside of Springfield.
Panda of Skunk Fu! knows what's best for Skunk (seeing that he's his master).
Louise in Slacker Cats is the voice of reason throughout the show but Buckley does zig-zag between being this and one of the many crazy people/cats throughout the show.
Stan. Kyle has the appearance of being sane as well, but unlike Stan, is unable to understand just how moronic the actions of all the insane people around are. Among the adults is/was Chef and, on occasion, Jesus.
In the "Pandemic" episodes, Craig took on this role, as he spends the episode calling out the main boys as being jerks that constantly get themselves and others into ridiculous situations, entirely in deadpan delivery.
Craig: "You guys took my birthday money, got me arrested and sent to Miami with no way home except to take down the country of Peru, and I'm being a dick?"
Cartman: "There's no talking to this guy."
Kyle's tendency to have quite a temper to the extent of being damn near bipolar excludes him from this on a regular basis, but there are instances where he plays this straight. The episode concerning the economy and the absurdity of aestheticism is a crowning example. Even "You're Getting Old" has him playing this to Stan of all people.
When he was around, Chef was usually one of the very few adults with any kind of common sense.
Sharon has become the closest to fulfilling this in Chef's absence.
President Bush, ironically enough, especially when compared to his cabinet and the press. "A Ladder to Heaven" contains a particularly infamous example.
By this point, it appears that Kenny is the only one that has showcased routine common sense throughout the series. This is arguably lampshaded by Butters (of all people) in "Going Native" where, after chewing out most of the others kids for their quirks, claims that Kenny is the only one with "any sense of dignity."
Apparently, Father Maxi is the only catholic priest who doesn't rape boys.
Plankton often shifts into this when dealing with Spongebob.
Despite being a pet, Gary also fills this role the vast majority of the time.
The titular character of Steven Universe must occasionally play this role when things get too emotional. This is lampshaded later on in the show, where his best friend Connie is pushed into the role during his absence and remarks in a short "The Reason You Suck" Speech how ridiculous it is that a child like her is more emotionally mature than aliens that are thousands of years old.
Titan Maximum deconstructs the idea of the Only Sane Man with Gibbs. Being the only competent member of the former Titan Maximum team eventually drove him into hating everyone else on the team and performing a Face–Heel Turn on the first episode. The fact that he's the Only Sane Man makes him a very effective villain, albeit an incredible Smug Snake.
Noah is more of a classic example, especially when you compare the rest of his team in World Tour (excluding Alejandro, who gets this whenever Noah's having fun).
Transformers Prime actually has this in one of its human characters, Jack Darby. Unlike almost every other human in the Transformers universe, he knows that getting yourself involved in the front lines of an alien robot war is more dangerous than awesome. In fact, he tried to avoid all contact with the Autobots at first before some convincing.
The villains have Soundwave, who doesn't share the sociopathy or megalomania of most of the other Decepticons and is far and away Megatron's most loyal and competent henchman. Be warned: This trope does not in any way make him less dangerous than the others. At all.
While Kitty from T.U.F.F. Puppy has the occasional tendency to act as goofy as everyone else, she is the most level-headed character when compared to Dudley, or anyone else on the show for that matter.
Belly Bag tends to play this role when Mr. Gus isn't there. He usually tries to rein in Uncle Grandpa's and Pizza Steve's inanities, although he's more likely to get caught up in it than Mr. Gus.
Bob the Tomato, in the VeggieTales cartoons — in the opening and closing sequences where he plays himself, he consistently provides a voice of reason on the show. During the actual episodes, he is almost invariably cast in the role of a character who is competent and sensible, unlike most of the others.
The Venture Bros. has Doctor Venture, who often tries to be the Only Sane Man, but his childhood always catches up to him. Usually, the role ends up being filled by one-shot bad guys that question the Venture Team's logic. You know you have problems when a flamboyant super-villain is judging you.
Brock fills this role quite often as well, in spite of murdering multiple people on an almost daily basis. He's generally the sanest, most level-headed and most competent member of Team Venture, so it's not hard for him to slide into the role and see just how absurd his life and the lives of those around him are.
Dr. Venture:(As a party being thrown by his brother sits on the brink of collapsing into madness) Huh. When you're not the one in the middle of it all, it's actually pretty obvious.
Brock: Welcome to my world.
On the villains' side, Dr. Girlfriend is likely the closest thing to a competent one (at least compared to the Monarch and anyone else affiliated with him).
In the X-Men animated series, the largest of the sentinels (robots designed to capture people) is called Master Mold. He/It starts trying to take over the world. When reminded by his creator that his mission was to protect humans from mutants, Master Mold says, "That is not logical. Mutants are humans. Therefore, humans must be protected from themselves." Not a bad realization for a robot antagonist that Professor X called "the embodiment of human hatred" that was created by another antagonist.