Linkara: This is ridiculous! There is no strategy here! Plus, he's my archnemesis! It's...annoying! Benzaie: No, they're right. We've got to fight madness with madness. And he's the craziest we got. (to Insano) Are you ready? Dr. Insano: (evil laugh) I was BORN ready!!! Linkara: (double facepalm)
The Gord, from the website Acts of Gord, apparently does this a lot. About every chapter of the "Books" has a small image of Gord facepalming while commenting on the worst idiocy on the page.
Topless Robot often features the page image in their "Fan Fiction Friday" articles. In one example, they have a combination of this and the image of Toht's face melting in a "Toht-Palm" image for passages in fan fiction that are disturbing and nonsensical.
There are giant montages available on certain image boards consisting of nothing but anime characters facepalming. There are montages of facepalm pictures that, once zoomed out enough, look like Picard facepalming. Such as this.◊
SF Debris: SF Debris very often talks about facepalming in his reviews of various sci-fi shows, focusing primarily on Star Trek. In the review of "Realm of Fear" from Star Trek: The Next Generation, he says he literally facepalmed when Deanna Troi counselled Reginald Barkley on his phobia.
In episode 3, Jane is absolutely desperate and touches and covers her face several times. She panics and freaks out because she's lost in the middle of nowhere at night, she lost her phone and can't contact the person who's supposed to give her a ride.
Jane facepalms at the beginning of episode 5 when she's exhausted and experiences "it's at the tip of my tongue" as she spent the whole day talking French with Adele, and cannot remember the word for "fatiguée". When she realizes it's "tired", she buries her face in her hands.
In episode 9 "Phlegm", Jane is ill and feels very down. She rubs her face and is on the brink of tears.
In episode 10, she touches her forehead in an exasperated gesture. It expresses how she's unsatisfied, lonely and restless. Also, it happened after an almost-collision with a car. People have a right to be upset after such experience.
In fact, the conclusion of every opening sequence, just before the title shows up, is one.
In the Color Dreams video (during the Raid 2020 part), he does a facepalm while wearing a giant Mickey Mouse-style glove.
At the end of the second special (about cartoons), the completely over-the-top scene from The Prince of Tennis elicits this.
On This Very Wiki, if a certain show/video game/comic/etc. has it's own Wall Banger or DMOS page, odds are an image of one of the characters facepalming will be the page image.
There's a "special feeling meme", in which two characters will be shown being interviewed for TV, with one saying "Being in the snow with your lover immerses me with a special feeling. I like it." while the other one blushes and face palms. A Puella Magi Madoka Magica example is here.
In Season 2 premiere of Noob an elite player turns out to be the younger brother of Sparadrap, who's so oblivious to his gaming activities that's he's convinced that he doesn't know how to play the fictional MMORPG in which most of the story is set. The scene revealing the family relation between the two ends with Sparadrap kicking his younger brother off the computer so he can play himself, noticing that the new update of the MMORPG got installed whitout him doing anything and... deciding it somehow happened by itself. His younger brother's reaction inevitably ends up as this trope.