Main page image shows Betty and Veronica of Archie Comics fame. The two started out the series back in the 1940s as bitter rivals, but some time around the 1950s were morphed into best friends / rivals in love. They still viciously fight each other over Archie, and occasionally play oneupsmanship, but for the most part (and Depending on the Writer) they're completely believable as close friends who just happen to share everything, including character flaws and boyfriends.
Jughead and Veronica bicker relentlessly, and seem to take glee in the other's failure. That said, they still come to the other's aid should things turn Dude, Not Funny!.
Reggie seems to have this relationship with everyone in the cast, which begs the question why they put up with him in the first place.
The aversion of this trope is one of the most noticeable deconstructions in Afterlife with Archie. Veronica and Betty absolutely hate each other, though when Veronica's mom died Betty comforted her. Veronica has Hidden Depths behind her aggressive personality but is constantly at odds with Betty.
Asterix himself usually has a bad argument with Obelix at least Once per Episode. It never takes long for them to make up, though.
Both of Fulliautomatix's main friends, Cacofonix and Unhygenix, seem to have this relationship with him. He relentlessly picks on Cacofonix, and he and Unhygenix both share hatred of his music (especially in The Falling Sky, where they are even more horrible to him than usual and get blamed for Cacofonix's hut burning down as a result). However, he and Cacofonix share mutual contempt for Unhygenix's merchandise—he's a fishmonger who sells stale fish despite living next to the sea. Fulliautomatix bullies both of them a lot, but also seems to see himself in charge of looking after Cacofonix—they're usually shown hanging around together in crowd scenes, he shows concern as well as satisfaction when Cacofonix loses his voice in The Roman Agent, he's genuinely torn apart when Cacofonix decides to leave the village in The Secret Weapon, and in one album he lets the Bound and Gagged Cacofonix sit next to him at the banquet at the end (although he probably tied him up in the first place).
Tony Chu and John Colby of Chew have this to an absurd degree.
Batman and Superman had somewhat evolved in this direction following the reinterpretation of Batman: The Dark Knight Returns, whose creator opined that the characters were too fundamentally different to ever be friends. Following the launch of Superman/Batman, they've been portrayed as good friends, but they are far from the World's Finest they used to be. Of course, Depending on the Writer, the relationship can be either "close as brothers" to a lukewarm friendship, and all the various points in between.
In those continuities where Lois Lanedoesn't know Clark Kent and Superman are the same person (for instance the Christopher Reeve films or Superman: The Animated Series) she is usually this to Clark, enjoying a very friendly workplace rivalry with him and affectionately chastising him for his timidity. In the very early Golden Age her dislike for Clark was genuine but their relationship quickly evolved into this trope. Later, more often in the post-crisis stories this went even further as her teasing covered up a massive crush on Clark.
Superboy: Turns out that the New 52 versions of Rose Wilson and Caitlin Fairchild are this, to the point that Fairchild left something behind for Rose to find if anything happened to her.
Batgirl and Spoiler are Heterosexual Life-Partners, but when they team up there is almost always good-natured grumbling on Stephanie's part, while Cassandra often playfully teases Stephanie about her comparatively inferior combat ability.
Lex Luthor and The Joker are a villainous example of this. No matter how often they fall out, no matter how often they come to blows, no matter how often they've tried to torture or kill one another they'll always end up working together again. Each has confessed to enjoying the other's company, and inasmuch as either is capable of having a friend, they view the other as someone who is fun to hang around with.
Tim Drake and Conner Kent have a downplayed version of this. Tim's interactions with Ives on the other hand are always full of snark.
Groo is oblivious to how much his sister Grooella hates him and thinks every insult and raging verbal assault shows how much difficulty she has showing her affection. But, then, Groo is oblivious to practically everything.
Shimy and Jadina from Les Légendaires. Due to the girls having opposite personalities and backstories, they constantly argue with each other, Shimy being easily irritated by Jadina's apparent silliness and shallowness while Jadina can't stand Shimy's Deadpan Snarker tendencies. Yet they both display extreme shock when seeing each other seemingly dead, and occasionally attempt to comfort each other. When Jadina learned Danael was attempting to have Shimy killed so Anathos couldn't get her body, she was enraged and yelled that she wouldn't let that happen. In the same vein, Shimy had a violent Heroic B.S.O.D. when finding what appeared to be Jadina's decayed body at the end of book 13.
The entire McKenzie farm animals cast from the Italian comic Lupo Alberto. Not only against the outsider Alberto, but also (and perhaps even more often) towards each other. The Berserk Button can involve unbridled gossip, the opportunity to become famous, uncontrolled fears, Glycerine the duck, complaints about work (which is generally tending the gardens)... heck, the list is long.
Santo (Rockslide) treats everyone like this, at least all the staff and students at the Xavier Institute. Anole, Pixie, and Mercury return it with the most obvious friendship.
Hellion. He's a smug, arrogant, self-entitled prick to everyone, including his friends (though less so with Mercury and X-23, with whom he's much less of smartass).
While they're not exactly best buds, Domino and Psylocke have serious shades of this in X-Force.
Domino: Hey, Princess? Shift a little ass, wouldja? We're s'posed to be prepping charges to blow this place. Y'creepy sanctimonious shut-in. Psylocke: Perish in a whirlwind of razors and lemon juice you unbearably perky irritation bomb. Domino: Good to see ya, Bets. Psylocke: You too, Dom.
Spidey arguably defines this trope with all the Anti Heroes he teams up with, especially with his no killing policy.
Wolverine and Spider-Man are a special case of an enduring friendship, arguably due to their moral alignments but also due to their methods in dealing with people. They butt heads, and fight each other over being two different flavors of effective hero. Still Wolverine trusts Spider-Man as one of the few heroes he would ever turn his back on; the others are Captain America and Charles Xavier.
The Torch also has a similar relationship with Spider-Man. Spidey and the Torch's team-ups usually consist of them trying to one-up each other, with the expected volley of insults and snark. But at the end of the day, Spider-Man is probably Johnny's closest superhero friend outside of the FF.
But all this pales in comparison to Spider-Man's unsolicited and unwanted "friendship" with Deadpool. Deadpool just loooooooves pressing Spidey's buttons, but will never hesitate to help clean up the messes he starts in Peter's life, and seems genuinely fond of the wall-crawler, sometimes even going out of his way just to hang out with him. Spidey, while often forced to admit that Deadpool does have his moments, more often wishes he would just go away.
Ben Grimm and Johnny Storm of the Fantastic Four spend most of their time either sniping and snarking at each other, planning increasingly elaborate practical jokes on each other, or lambasting each other with creatively brutal threats about what they'll do to the other (which are hardly ever followed through) after said practical jokes. They are, of course, practically inseparable.
Cable and Deadpool become this more and more throughout their series.
Before that, Deadpool had this relationship with Bullseye. They took it to the extreme though, where they would try to kill each other and then laugh about it together. When it's all said and done, Bullseye has admitted that Deadpool is the only person he likes.
Deadpool and Taskmaster have also had this type of "friendship"—perhaps said best by Taskmaster: "The things I do for the friends I can't stand..."
The Defenders, in most incarnations. They are in every way a "non-team" of "non-friends," and the standard Defenders' parting words are "Let's never do this again," but if one of them comes to the group for help, s/he'll get it.
Michael Morbius and Jack Russell have been friends for a good few decades — a friendship filled with sneering, physical fights and death threats. Still, when it comes to it they won't stop at anything to save each other, sharing a kinship in their respective curses that very few others would understand.
Doctor Doom and Namor the Sub-Mariner. They're both pompous blowhard monarchs who can tolerate very few other people on the planet, and they fight half of the time in the comic series Supervillain Team-Up, but they respect each other as monarchs, and powerful individuals who make hard decisions whereas most people both of them deal with don't rule nations. The only person that comes close to their position is Black Panther but he doesn't share their arrogance.
Victor and Frank from Mini Monsters. As they're polar opposites, both spend their time insulting and bickering with each other constantly. However, neither of them can be separated and they're best friends, although Victor denies this.
As children, Sam and Max (of Sam & Max: Freelance Police fame) seemed to be this way. Max used to mercilessly make fun of Sam, who was too shy to ever stand up for himself, but wouldn't stand for anyone else doing the same.
Transformers: More than Meets the Eye: Ratchet and Drift. Ratchet spends most of his time making sarcastic comments about Drift, who (usually) just shrugs it all off, but underneath it becomes apparent Ratchet does actually care about what happens to Drift. Just don't expect him to ever admit it.
Transmetropolitan: Spider has this type of relationship with pretty much everyone who can even stand him (the two exceptions being Vita Severn and Oscar Rossini), but his relationship with Mitchell Royce stands out even for him. Royce has held Spider at gunpoint to force him to work (Royce describes it as "going in with a gun and coming out with either a story or a corpse"), and Spider has at one point gotten Royce infected with an STD that made his testicles sing "Twinkle Twinkle, Little Star". However, Royce has put his career on the line for to run interference for Spider's sake more than once, and Spider is genuinely appreciative. When Spider has to "vanish", and has reason to believe he will never see Royce again, his last words are:
Goodbye, old friend. You always did right by me.
Eric and Woody from Quantum and Woody, two lifelong friends who spend more time fighting each other than the bad guys. The fact that they have to meet every 24 hours to touch their quantum bands just adds to the vitriol.