The very first one in Star Trek: The Original Series happens in the pilot episode with Captain Pike when he tries keeping up a steady stream of hateful, violent thoughts as Psychic Static. Vina, who'd been living with the Talosians for twenty years and told him she'd tried that more than once, just puts her hand to her face when the Talosians proceeded to give him the mental agony beam.
Captain Picard often does this, as the Trope Image from "Deja Q" shows. The image of Picard has become the patron saint of facepalming. The alternative "Single Head Double Facepalm" was featured in "The Offspring".
Commander Riker had a memorable facepalm in "A Matter of Perspective", the very next episode after "Deja Q", humorously enough, seen here◊.
There's a meme image where, in the Riker image above, they photoshop the Picard facepalm for a "double facepalm".
Star Trek: Deep Space Nine. Sisko is often fond of the pinching the bridge of ones nose variation of this trope. Kai Winn is most often the cause. But he lets out a few outright facepalms here and there.
"Death Wish", Q performs an elusively rare variation on the double facepalm. That is, two faces, one right palm per face◊ (this is not a recommended procedure for mere mortals). Oh, whatever was the gesture for? Well, Vulcan logic is no fun, basically.
In "Tinker Tenor Doctor Spy", the Doctor was posing as captain while aliens threatened attack. When he started improvising his lines, Janeway facepalmed.
Kamen Rider Kabuto. In the Hyper Battle Video, Arata Kagami attempts to perform a Rider Kick using Kabuto's style. Tendou Souji couldn't bear to watch the Epic Fail that ensues.
Giles, mentor for Buffy the Vampire Slayer, often takes off his glasses and squeezes the bridge of his nose when confronted with the thought patterns of teenagers. He also tends to clean his glasses with a handkerchief a lot.
Buffy: Is that why you're always cleaning your glasses? So you don't have to see what we're doing? Giles: Tell no one.
One of Jack Benny's signature bits was to look at the camera with a hand to his cheek and a pained expression on his face.
A scene from Too Close for Comfort episode "The Remaking of Monroe" has a co-worker say he is surprised that Henry is 52 years old (he thought he was two years older). Henry then asks Monroe (who just arrived at Henry and Muriel's apartment a couple of minutes before to ask for money to attend an assertiveness training seminar) how old he looked. Muriel (who is standing behind Monroe) mimes to him a response by holding up four fingers on her left hand and five on her right. You would think the response he would go with based on Muriel's hand signals would be "45", but instead Monroe goes with the response, "nine". What Muriel does is actually more of a head palm, but still...
Al Bundy sometimes does this on Married... with Children when life was acting particularly cruel to him. He also does the head-desk variation in one episode when he bashes his head against the table in frustration while trying to pay some bills.
In Farscape, the mercenary Bekhesh performs a faceplatepalm when Stark explains that the plan to attack the Shadow Depository relies on a silent count — in the middle of a potential shootout with the guards.
President Bartlet from The West Wing does the head desk variation on the frickin' Resolute desk while stuck listening an Old Soldier Old Diplomat's "fascinating" stories about various Cold War-era foreign relations escapades.
In The Big Bang Theory, Leonard probably does the longest facepalm ever televised when the science panel they are supposed to be speaking on deteriorates into complete and utter chaos as Howard and Bernadette start comparing Howard's... umm... weaponry to one of her ex's. Once he realizes that nothing will save the situation, he joins the "fun".
Spencer balls his fist onto his face when the European swimsuit models dance goofily.
Carly slaps her forehead with her palm upon learning Sam's mom has no idea what iCarly is.
USA has a commercial for both Burn Notice and White Collar that has Peter Burke noticing Fiona's gun that she has taken out of her purse at lunch to answer her cell phone. When she tries to cover the gun with her purse, a grenade rolls out of the purse. Michael's facepalm is almost as epic as the trope image.
Bert Convy did this in embarrassment whenever he blurts out the clue word or solving the password in Super Password.
In Glee, Kurt has never gone as far as an overt facepalm, but in moments of intense exasperation will press two fingers hard against his temple. It's the understatedness of the gesture that makes it so effective.
This was played with in Criminal Minds, where the team leader does a face palm after the team geek makes a particularly dorky statement. However, it turns out that he's experiencing a headache due to an injury sustained in the previous episode (where he was blown up by a car bomb).
Evelyn does this in Two and a Half Men at Charlie's funeral when Rose describes how he wasn't in pain at his death (his body merely exploded like a meat sack when that train hit him).
On Boy Meets World, Shawn and Minkus both facepalm in unison when Mr. Feeny assigns them as partners.
The father on Family Ties is reduced to a face whipe version of this and borderline angrish when he came home early to find a wild college party going on the house.
"There was... a kangaroo... in my house..."
On NUMB3RS, Charlie did this once when Larry said that perhaps the suspect could be in two places at the same time. (It was a quantum idea.)
There's a sketch from It's Marty in which a terrible elderly couple utterly demolish a travel agent.
Travel Agent: Would you like to travel by plane, train, or coach? Couple: No. Travel Agent: ... No, what? Couple: No, thank you. Travel Agent: That's bett—(facepalm)
In The Mentalist, Teresa Lisbon, gobsmacked by the antics of her consultant Patrick Jane, teeters on the edge of executing one for seasons 1 and 2, before she finds out in Season 3 that Jane broke into a suspect's house for the information they used to arrest him — in the middle of the suspect's trial — resulting in a beautiful example of the trope.◊
Inspector Lestrade has a truly epic one after Sherlock pulls a gun on a group of police officers, pretends to take John hostage, and the pair take off down the street while still handcuffed together.
John also did one when he found a head in the fridge. Later on, he did it again during an incredibly awkward meeting between Molly, Jim and Sherlock himself. Molly is trying to provoke Sherlock into jealousy. Jim is unashamedly fawning over Sherlock. Sherlock is being obnoxiously rude to both. And everyone is ignoring John. Good times.
On Get Smart, the Chief gets "headaches", grasping the bridge of his nose with his fingers, whenever Maxwell Smart does something particularly stupid.
In one post-Channel Hop episode, Max does the palm-in-face version when 99 is shown to have acquired some of his habits.
The X-Files: Over its nine-year-long run, the show had several facepalms of various flavours ("Am I the last sane person here?", "I can't believe I did something this stupid", "All right, what now, any other crazy idea?", "Oh no, not this", "Oh no, not this again!", "Could this be any more humiliating?" or "I might as well break down completely.").
"Syzygy": Mulder rubs his face with a hand when Detective Angela White comes to his motel room and he's not sure what's going on exactly.
"Bad Blood": Scully facepalms when Mulder wants them to compare their accounts of one curious case that involved Mulder sticking a stake into a teen's chest. The facepalm followed his Prison Rape joke about Scully' hypothetical prison mate nicknamed Large Marge.
"Patient X": Mulder is among panelists at a UFO conference. Other speakers take the abductions for granted and they enthusiastically talk about "the ontological shock" that must follow. Mulder is looking down, his hands are tightly joined and his thumbs touch his nose bridge. He's deep in thought, trying to strengthen himself... because he's about to call it all bullshit and blame it on The Government as their conspiracy.
"The Red and the Black": At the end, when Mulder is arrested by some military men and thrown to a car with Scully, he's tearful and he sadly covers his face. Scully touches his hands and tries to comfort him.
"Rain King": Scully facepalms when Mulder suggests that Holman Hardt is manufacturing the weather. The facepalm is complete with her "look".
"X-Cops": Scully holds her mouth and turns away from a camera when she finds out it's rolling for TV.
"Hollywood AD": Mulder and Scully are in the theatre, watching a film based on one of their cases and their characters. The audience chuckle, the actors look interested and pleased, AD Skinner looks happy. Scully watches the film in horror, but her shock and surprise is smaller than Mulder's, who just drops his head into his hands and hides his face completely.
"Hollywood AD": When Skinner tells Mulder that his friend from college, now a writer/producer, will shadow Mulder and Scully on their case, Mulder just rubs his forehead and asks if he pissed off Skinner more than usual.
"Je Souhaite": Scully puts her face in both her hands because she invited some scientists from Harvard to see an invisible body which unfortunately disappeared. She's beyond embarrassed.
"The Truth": Agent Doggett tells Scully that Mulder got death by lethal injection and understandably, Scully just loses it and heart-breaking sobs escape her uncontrollably. She covers her face with both her hands and continues to cry.
Horatio Hornblower: Horatio facepalms in "The Duchess and the Devil" when he learns that some men of his crew plan an escape from their imprisonment. He wants to escape as well, but they want to leave Archie behind and it's all too clear to Horatio that their hasty plan cannot succeed.
The Misadventures of Sheriff Lobo. The title character does this at the end of the Title Sequence when all his deputies crash their squad cars into each other after Sheriff Lobo tells them to "Move 'em out!"
When the financially hopeless Lord Grantham is talking about investing his fortune on Downton Abbey, he mentions that there are schemes out there that offer a double or triple return on investment - "There's a chap in America, what's his name, Charles...Charles Ponzi who offers a huge return after ninety days." Matthew can be seen facepalming.
Friends: The Friend most likely to be reduced to sinking his face into the palm of his hand at the insanity of the rest of the gang is Chandler, who resorts to it often. It goes along with his Deadpan Snarker role in the group. However, the others get their moments, too, and Chandler's also been the cause of some on top of that.
Red Dwarf: Both Lister and Rimmer frequently facepalmed because of each other's insane ideas, general smeg-headedness or extreme and disgusting slobbishness.
Ally McBeal: Many characters many times, usually when they screwed up at court or at the meeting. Very often it was Richard Fish who caused his colleagues to bury their faces in their hands. Sometimes they got to the level of synchronized triple Face Palm.
Columbo: Lt. Columbo would do it quite a lot when something obvious would suddenly occur to him.
Pee-wee's Playhouse. In the episode "Stolen Apples", after Mrs. Steve tells Pee-wee that Randy has been stealin her crab apples again, Pee-wee facepalms himself.
Monday Mornings: Viewers get to see characters facepalming all the time, both for their own failures or because others failed spectacularly. It might be expected in a Medical Drama dealing with professional screw-ups. They are seen especially during their M&M [Mortality and Morbidity] meetings, but there are several during legal meetings or after surgeries. Some Face Palms were related to shame warped by personal relationships.
Jen or Roy would often facepalm to express their frustration or humiliation.
When Roy and Moss run on a bomb disposal robot, the whole company watches them from a safe place and everyone facepalms epically.
In the Czechoslovakian sci-fi series for kids The Visitors (1983), the forehead slapping variant of Face Palm became a running joke. When the group of scientists from the 25th century first came to the 20th century, they stopped for a group of hitch-hikers. The guys were happy at first, but seeing that the car was completely full, one of them just slapped his forehead in silence. Professor Richard concluded that it was some kind of greeting and he started using it to everyone's utter confusion. Eternal Czech was not even an issue and communication was more or less ok, but something as universal as Face Palm is apparently unknown on Earth in this vision of blissful and very much Star-Trek-esque future.
In the drama Wizards of Waverly Place (2007-2012) Justin uses this a lot. And everyone has used this at least once when it comes to Max.
In the pilot episode of Malcolm in the Middle, Malcolm double-facepalms in exasperation and humiliation after his teacher stands him in front of his class to patronizingly tell them how he's "very different, in his brain!" from his classmates, after he is revealed to be a genius. He is still facepalming in the next scene when he's in the middle of his new gifted-section classroom, then gingerly peers through his fingers, providing the clip used as the title card of every subsequent episode.
René in 'Allo 'Allo! facepalms frequently when Michelle of the Resistance introduces another of her brilliant plans, when his wife sings, when Monsieur Leclerc messes up or shows up in his Paper-Thin Disguise, when malaproperstastic Officer Crabtree opens his mouth, when Camp Gay Lieutenant Grubers hits on him. Basically at least once an episode, and other characters might join him as well.
A sketch from The Benny Hill Show combines this with physical comedy when one character (Benny Hill), upon seeing his friend (also Benny Hill) with a huge bloody hole in his forehead, asks him how he came to get it. He answers that he had recently been in the company of a woman but failed to realize her very obvious signals... and, upon doing so much later, put his hand to his forehead — forgetting he had a sharp piece of equipment in his palm at the time. The funny part is, upon demonstrating what he did to his friend, while holding the pointed bit of metal, he manages to put a second hole on the other side of his brow.
In the very first episode of Zorro, Commander Monasterio (while stuck inside a prison cell) facepalms as he watches Zorro utterly humiliates Sergeant García and the rest of his soldiers for the first time (but certainly not the last).
In "Lazy Scouts" on LazyTown, as a Funny Background Event during a good portion of the "Lazy Scouts" song, Stephanie is standing in the background, doing a double facepalm, rolling her eyes, putting her hands on her hips, shaking her head, or otherwise expressing disgust.