Doom Patrol featured the Brotherhood of Dada. They were pretty darn weird, and considering that this comic is the example of What Do You Mean, It Wasn't Made on Drugs?, that's saying something. One of their most memorable moments was when they first met the Doom Patrol and the leader of the Brotherhood, Mr. Nobody, threw a fish on the ground and said "There! Now we've taken over the world! What are you going to do about that?"
In the German comic Werner: Some one-pagers can only described as that.
"The Angriest Dog in the World" consists of a panel that briefly explains the, um, "premise" of the comic, followed by three panels of the dog in his owners' yard. The final panel is the same as the preceding three, but set at night. The only way to distinguish one comic from another is by the word balloons emanating from the house, which expound on topics ranging from silly puns to existentialist dilemmas to the makeup of obscure chemical compounds. Oh, by the way, it's written by David Lynch.
Gary Larson's The Far Side (no recurring characters, with some minor, still non-canon exceptions), twisted non-sequiturs, blenderized pop-culture in-jokes — pretty much the forerunner of all the other Dada Comics in the newspapers)
Edward Gorey's blatantly surreal and occasionally nightmarish illustrated short stories provide an Ur Example for this trope, with particular credit due to The West Wing, The Epileptic Bicycle and [Untitled].
Pokey the Penguin, a bizarre comic about a penguin who lives in the Arctic Circle with Mr. Nutty, a drunken and inexplicably British snowman, Skeptopotamous, an Eeyore-esque hippo, and a bunch of other penguins, going on crazy misadventures and occasionally butting heads with the Italians, who want his Arctic Circle Candy. Dialogue is in all-caps, occasionally scribbled out or in strike-through, and is peppered with lines like "THE FLYER HAS A CUTE KITTEN DRAWN ON IT! IT REPRESENTS VIOLENCE AND CARNAGE."
Chainsawsuit by Kris Straub, of Checkerboard Nightmare. Almost none of the comics are related, except for some recurring characters such as Two Cops, the cop who enrolled in the Police Academy twice by mistake, or Cthulhu. Found here.
If Listening To 11.975 MHz doesn't count, nothing does. It seems to take place in a literal Dream Land. The cast includes a girl with antennae (of Pac-Man chasing a ghost) who speaks only nonsensical French, a girl in swirly Opaque Nerd Glasses who speaks only in calculus equations, a Chinese Girl who speaks only nonsensical Chinese, a Barbie Doll Anatomy-subverting Innocent Fanservice Girl (possibly Brazilian, from her dark hair and Markov-chain Portuguese) a Hippie Chick who wears 1 less item of clothing each time she appears (last seen wearing a lab coat, glasses, peace sign, hair band, sandals, and nothing else), and a walking radio that can only say the words "Zachary", "Acetaminophen", and "Beige". Oh, and random scenery that quotes... random things. Like the names of European dictators, or random snippets from 1960s novels. Randomly. This doesn't include the backgrounds and minor characters, which can be charitably described as "screwed up." Oh, and the contact page gives information on contacting the author over CB radio.
(Note: The author claims to have never heard of Dadaism.)
Super Mega Comics certainly qualifies. The art consists entirely of poorly-drawn stick figures that put Shirt Guy Dom to shame, and the plots make a marginal amount of sense, at most.
Witch's Brew is literally Dada. Each strip is produced via the Exquisite Corpse game, where multiple artists collaborate on each strip, but must make their contributions without seeing any more than one panel of the strip.
Chicanery, a sprite comic starring an assortment of Earthbound characters and other clip art. After the events of the game, Pokey embezzles $44 million from Giygas so that he and Ness "have enough legal tender to go on nonsensical zany adventures". They are quickly joined in their escapades by a trigger-happy Mr. Saturn who is relatively more coherent than others of his kind, Mr. T (represented by the "generic black guy" sprite from Earthbound, which looked unmistakably like the real T), Ness's former comrade Jeff, and Pip from Chrono Cross. These escapades include traveling through time to prevent a nuclear detonation (and briefly ending up in medieval times, represented by graphics from Ultima I), the occasional treasure hunt, extended parodies of Parasite Eve and Metal Gear, and a fight between a giant Mister B. Natural and a Humongous Mecha based on MST3K's Frank Conniff ("the 2000-inch TV's Frank").
Beaver and Steve, in their unfeasible adventures, do encounter the occasional recurring character or plot point (which occasions probably make for the weaker of the strips); at its best, though, this sometimes-single-page, sometimes-story-arc webcomic shows no interest in consorting with such dull companions as continuity or logic. There Are No Rules here. Anything can happen. (Although there is a better-than-average chance that there will be toasters, robots, and time travel involved.)
Framed!!! is very surreal, starting with its self-referential metafictional premise of real people trapped in a comic strip. It frequently identifies conventions just so that it can violate them, such as having a character turn into abstract art or shove a tilting panel back into position. It's also responsible for the massive "Framed!!! Great Escape" crossover event, a weeks-long orgy of metafiction.
Loserz sometimes falls into that, as in this strip.
And Garkov is what happens when you throw the dialogue in a blender.
The Dada Detective, which neatly subverts this trope by having a technically sensible plot — a detective is trying to find a missing duck — but is filled with talkative mimes, goons who are overly fond of metaphor, disco, and Peter Lorre.
mezzacotta. It's expected when there are trillions of strips and that they are generated by randomizing the lines.
Slow Wave is a collective dream diary in comic form. It makes roughly no sense. Although this might be a subversion, since A) The whole thing is descriptions of dreams, and B) This Troper sent in a dream which the author rearranged to make less sense.
megaGAMERZ 3l33T is a parody of this genre. It is ostensibly written by Diablo, a fictional evil chicken (not to be confused with all those real evil chickens) from the comic Goats. Early strips are barely distinguishable from real Dada Comics, but over the comic's year-long run "Diablo" pushed the comic into being a more obvious and extreme parody.
This one's called Dada. There are references to the original Dada movement; for instance, the Mona Lisa is a major character. Plots are followable, but are interrupted by word-association battles so often that you might not notice.
Sweet Bro and Hella Jeff a webcomic that ties into Homestuck. It's a two-gamers-on-a-couch series only totally incomprehensible and so ironic you can't even begin to understand all the layers of irony.
The Book of Biff. Every single update, Biff has found some way to defy the laws of physics, do something inexplicable, or simply apply everyday objects in absolutely insane ways. The second comic pretty much solidified this early on — it involved Biff inexplicably trying to smother a grease fire by beating it with two hammers. (This has since become a mini-meme among the commenters for early comics.)
This random generator provides a new Dada comic every time you refresh the page. WHEN ARE METEORIC INTERSCINDINGS MARLING YOUR FORWARD PITYINGS?
Fiascos is a comic about an unnamed man in a suit with a TV for a head and a fluorescent alien called Jam try to return to their homeland. Things descend into chaos almost immediately.
Educomix, despite picking up a storyline a few strips in, remains completely insane in comparison to anything in real life, yet runs on its own consistent internal logic.
keithiscoolbykeith was a mind-bogglingly random.lo-rez and deliberately(?) amateurish strip. And then it randomly disappeared.
''Negamaki!'' starts with talking sushi, immediately devolves to Cthulu Mythos, and pretty much keeps riding the slide down from there.
Ghettonauts, a fancomic of Psychonauts, has it's first page being a shot of one of the campers laying in the grass, apparently on an acid trip, and that's it. It gets weirder.
Deer Me has an occasional example with in-verse comic "Wonder Wonder" wherein the featured Wombat does nothing except say one word in the final panel. Viana loves it while the rest of the cast understandably fails to see the appeal.