Stan is a big fan of the "pinching the bridge of one's nose" type. It makes sense, as, thanks to Flanderization, he has a Liz Lemon Job. Often accompanied by a muttered "Oh goddamnit..." Apparently this is because series co-creator Trey Parker (on whom Stan is based) does this a lot in Real Life.
In one episode, we see his mother doing it along side with him, implying that he picked it up from her. Given to who she's married to, that may, very well, be the case.
Cartman does this in the third part of the Black Friday trilogy when Princess Kenny breaks up a blockade of ships trying to keep a shipment of PS4s out of America just by being cute.
Sadlygrove often inspires this to his friends — usually along with calling him a "Iop brain". Eva goes for the two-handed facepalm◊ in episode 19 when Grovy gets himself into a duel with Prince Armand.
Jay the Iop also spurs Kriss Krass to facepalm during the Bontarian Gobbowl match.
Rushu facepalms when he thinks Anathar has just crushed his precious Eliatrope prisoner.
In an episode of Batman Beyond, Batman (Terry McGinnis, remember) performs the motion after his geeky schoolmate Howard tells his obsessive-compulsive robot girlfriend they could be just friends.
Howard: Wrong thing to say? Batman: Is it ever the right thing to say?
An episode of Batman: The Animated Series has Veronica Vreeland attending the opera with Penguin. Imagine her reaction when the Penguin starts singing along.
Toa Tahu does this in BIONICLE: Mask of Light during a Kohlii match after Jaller dives right out of the way rather than blocking the oncoming Kohlii ball. Amusingly, shortly after, Jaller hits himself in the head with his Kohlii stick. It's an incredibly dramatic double-facepalm.
Squidward does this occasionally, and will add to it by sliding his tentacle slowly off his face, causing his face to sag and then spring back into shape.
Plankton does this after telling SpongeBob to be more assertive and SpongeBob misunderstands.
Mac Mange, the Metallikat of SWAT Kats, has a metal face palm in "A Bright and Shiny Future" when his wife (with whom he co-ruled the city in a Bad Future) kept insisting that their robot servants were hers. Complete with metal clang.
Twilight's typical response to the strange happenings around her, though some other ponies get one in now and then, too.
The other Ponies get in on it too, with Rainbow Dash doing a Double Facehoof at times. Behold fanart of a quadruplefacehoof.
Angel Bunny does a facepaw whenever he finds Fluttershy frustrating.
One of the Diamond Dogs facepaws when Rarity is aggravating him.
Spike faceclaws from time to time.
In "Inspiration Manifestation", Owlowiscious simulates this with his flight feathers when Spike reads the cover page of the book and says "I'm likin' the looks of this one!" Even more impressive in that he used one of his wings to do it while in mid-flight, and still remained airborne.
In "Equestria Games", Spike's cringe-worthy attempt at the Cloudsdale anthem naturally induces this to much of the audience (both in and out of universe).
Bloom & Gloom": This is Octavia's reaction in Sweetie Belle's dream when the latter gets a broom-and-bucket cutie mark. Then Rarity does a double-facehoof when she has to score her sister's performance.
Now compiled in this handy compilation! Begins at 2:45.
For the background ponies, this seems to be Bon-Bon's default reaction to Lyra's antics when portrayed by the fandom. Octavia and Vinyl Scratch get in on it whenever the other tries to listen to their preferred music genre. In fact, the only real common pairing of ponies that doesn't have a facehoof specializer (regularly, at least) is Derpy and Doctor Whooves. Probably because they both have their antics at the same time.
Doktor Frogg does this quite a lot in League of Super Evil when Voltar comes up with ridiculous and childish "evil" schemes. This is actually quite a talent, considering he doesn't actually have hands....
Looney Tunes, Duck, Rabbit, Duck! Elmer goes gunning for Bugs, and blasts a snowman-rabbit to pieces. Bugs floats down in a poor-quality angel costume, and Fudd falls for it. Cue angry, exasperated facepalm from Daffy.
The eponymous turtle has done this at a least a couple of times in the various animated incarnations of Franklin.
In Teenage Mutant Ninja Turtles (2012), Donatello facepalms when Michelangelo remarks on what a coincidence it is that a van has crashed into the Kraang's lair whilst the Turtles are sneaking in. Unfortunatley, Donny is wearing spiked climbing-kncukles at the time. Ouch!
Facewing by Connie Crane in "I'll Be Your Best Friend" on PB&J Otter when her husband falls into the lake after craning to get a look at Bucky Spacebeaver's showboat; Flick later on in the same story after Munchy brings Peanut a chocolate-and-tuna flavored cake and he realizes he's just been outclassed.
Ned from Ned's Newt does this if Newton is acting unintelligible to him.
An example is in "Broken Record" where Newton gets the idea to jump off a cow in an attempt to break a world record for Ned, much to the latter's disappointment.
In Xiaolin Showdown Chase Young uses one of the magical Shen Gong Wu objects to talk to animals and tell an army of monkeys to obey the every command of Jack Spicer. He facepalms when Jack tests the monkeys' loyalty by having them all do the robot dance.
In The Powerpuff Girls episode "Meet the Beat-alls", Princess does this in response to Mojo Jojo's lame pun
Princess: So tell us again why we're only stealing toilet paper, light bulbs, milk, flour, and eggs. Mojo: Well, Moko's idea is that stealing items that are all white isn't against the law. So it's okay to take them. Princess: Huh? Mojo: It's alright, 'cause they're all white!
The Dreamstone: Big Bad Zordrak does this fairly often in response to the incompetence of his mooks, the Urpneys.
On Wonder Pets, Ming-Ming does a facewing in "Help the Easter Bunny" after Linny asks what the song and dance routine he just made the group do had to with hiding eggs and he replies with "What eggs?" This episode being late in the show's run and her being the Little Miss Snarker of the group, it's probably not the first time she did it.
On Toot & Puddle, Puddle does it in "Desmond's First Snow" when Toot says he can feel for sure in his ears that it's going to snow after already having been wrong once before.
On the PBS animated version of The Berenstain Bears, this is Mama Bear's reaction in "The Green-Eyed Monster" from the PBS series when she explains the Green-Eyed Monster to Sister Bear after she wants to ride Brother's bike and she just whines "But Mama, it's such a beautiful bike."
On Holly Hobbie and Friends: Christmas Wishes, this is Carrie's reaction when the Hey Girls are coming up with ideas to help the Deegans have some Christmas cheernote The father of the family has recently passed away. and Amy suggests that they could organize their sock drawer.
Kaeloo: Mr. Cat does this quite a lot. In Episode 105, he uses the "pinching the bridge of the nose" variant of it several times.
Kaeloo herself does this on occasion when Stumpy does something stupid.