Main page image shows Betty and Veronica of Archie Comics fame. The two started out the series back in the '40s as bitter rivals, but some time around the 50s 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 together, 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 anyway.
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.
Then again, when facing a resurrected Joker, Terry disabuses him of this notion quite thoroughly.
Terry: The real Batman never talked to you much, did he? That's probably why you were so fixated on him. [...] The real reason you kept coming back was you never got a laugh out of the old man. [...] Get a clue, clowny! He's got no sense of humor! He wouldn't know a good joke if it bit him in the cape! ... Not that you ever had a good joke.
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, canon has them 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.
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 special case of an enduring friendship, arguably due to their moral alignments but due to their methods in dealing with people. They buttheads, 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 never 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.
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 as well. 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).
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..."
Victor and Frank from Mini Monsters. As they're polar opposites, both spend their time insulting and bickering each other constantly. However, neither of them can be separated and they're best friends, although Victor denies this.
Asterix himself usually has a bad argument with Oblelix 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).
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.
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.
Shimy and Jadina from Les Légendaires. Due to both 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 BSOD when finding what appeared to be Jadina's decayed body at the end of book 13
Michael Morbius and Jack Russell have been friends for a good few decades — a friendship filled with sneering, psychical 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.
Tony Chu and John Colby of Chew have this to an absurd degree.
In those continuities were 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.
The entire Mc Kenzie farm's animals cast from italian comic Lupo Alberto. Not only against the outsider Alberto, but also (and perhaps even more often) towards themselves. The Berserk Button can involve unbridled gossip, opportunity to become famous, uncontrolled fears, Glycerine the duck, complain on work (which is generally treat the gardens)... heck, the list is long.
Superboy: Turns out that the New52 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.