Superman and Batman 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. That said, in recent years it's definitely leaned much more towards the former, with it being pretty much uncontested that Superman is Batman's best friend and vice-versa.
In those continuities where Lois Lane doesn't know Clark Kent and Superman are the same person 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 Of Comic Books her dislike for Clark was genuine but their relationship quickly evolved into belligerent friendship. Later, this went even further as her teasing covered up a massive crush on Clark.
Supergirl and Thara "Flamebird" Ak-Var used to be like sisters until the manipulations of the villains during the New Krypton and Who is Superwoman? storylines caused them to have a fallout. Even so, Chris "Nightwing" Kent thinks Kara and Thara just don't want to admit they still love each other. So, when Lois Lane wonders in The Hunt for Reactron why Kara and Thara seem to be constantly fighting, Chris answers they are good friends. He would later be proved right.
Lois Lane: Thara is with Kara right now? Nightwing: Should be. Lois: Is there a reason every time I turn on the news they're beating on each other? Nightwing: Uhmm. I think it's because they're best friends? Lois: Ah.
Lex Luthor and the Joker are a villainous example. 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.
Robin Tim Drake and Superboy Conner Kent have a downplayed version; though Kon gets really snarky with Tim if Tim asks for his help without having a clear picture of the situation, it's still good-natured ribbing. Tim's interactions with Ives on the other hand are always full of snark.
Damian Wayne and Jonathan Kent are played up to this heavily, even more so than their dads.
In Superboy (New 52), Rose Wilson and Caitlin Fairchild are this, to the point that Fairchild left something behind for Rose to find if anything happened to her.
Depending on the Writer, The Joker either imagines that he and Batman have an extreme variation on the trope, or just pretends he does because it bugs Batman. Considering what he did to Batman, Joker is really lucky Bats has a no kill policy.
Batgirl: Cassandra Cain and Stephanie Brown 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.
Barry Allen aka The Flash and Hal Jordan aka Green Lantern. Sure, they argue and bicker at times, but the rest of the time they joke around like old friends. The New 52 and Blackest Night comics especially have Barry and Hal teamed together with the same gusto as Superman and Batman.
Big Barda of the New Gods had this relationship with her husband Mr. Miracle's Best Friend Oberon. The two frequently teased and insulted each other — Oberon declaring that Barda was all legs and no brain, Barda mocking Oberon's diminutive stature — but were good friends even if they'd never admit it.
