Superman is basically the Trope Maker for the entire genre of modern superhero comic books. While there were super-powered characters before him, none of them included all of what are now seen as the "classic" Comic Book Tropes the way Superman did. Every superhero ever written since basically follows in his footsteps.
The Golden Age of Comic Books also had a secondary trend of vigilantes with ill-defined omnipotent powers, who'd fight street crime without much regard for ethics. See Dr. Fate, Green Lantern, the Spectre, Black Widow, and so on.
In the Silver Age, all comics followed trends, to the point where it was an in-joke among comic writers and fans, for example an EC story about hiring Jack Kamen includes the line, "Jack, you old son! I haven't seen you for two trends!"
EC of course, followed trends religiously for a while, then started their own, Horror Comics! This in turn led in part to the Comics Code (boo! hiss!).
Marvel Comics' success, particularly with Spider-Man, the first teen superhero, had many publishers trying for a more teenager-friendly product; sadly, these often faded into Totally Radical.
On the subject of Spider-Man, his comics were also one of the first to illustrate the "normal" side of the hero, along with Fantastic Four. Rather than filling the issue with one action sequence after another, part of the issue would illustrate Peter taking on everyday tasks such as getting to work on time, experiencing relationships, dealing with school bullies, and so on. Even nowadays, polls and streets interviews indicate that the main reason people like Spidey so much is because "he's a regular guy like the rest of us." It has since become standard for comics to portray the everyday side of the superhero, with the character, like Peter, being portrayed as someone the target audience can relate to. Unfortunately, under worse authors, this often results in myriad forms of This Loser Is You.
Before "All New All Different X-Men" superheroes were WASPs.
And it's even Older Than They Think, the first international interracial hero team was Cyborg 009.
Nobody who's written Batman in the past 20 years has been able to escape the influence of The Dark Knight Returns. This case is particularly hilarious because the single greatest influence on Batman's character wasn't even canon. Even Batman's entry in the All-Star series, which was supposed to throw out all the complicated backstory and let the DC heroes have more Silver-Age-style adventures, was written by The Goddamn Frank Miller himself, and Batman was even more cranky and psychotic than ever.
One especially influential aspect of The Dark Knight Returns is the blow-by-blow First Person narration. Apparently inspired by Taxi Driver it was distinctive when Miller first did it. Then all Batman comics had it. Then just about all of DC. Marvel soon followed suit. Now it's pretty much universal in mainstream comics, whether it suits the book's tone or not.
The ever-hilarious YouTube series Marvel/DC Happy Hour parodied this by having the second season's Story Arc involve the Joker using nanotechnology in an android named Lance to brainwash heroes like Superman, Wonder Woman, Iron Man, The Incredible Hulk, and Wolverine into thinking that their loved ones died horribly right in front of them so that they would be more dark and brooding, like Batman, except that they would break his one rule in pursuit of Justice. The only heroes he couldn't brainwash were Ghost Rider, The Punisher (because he was already a dark, brooding, homicidal vigilante), and Spider-Man (because the events of One More Day had already altered his perception of reality, and he has lost everything but never loses his optimism).
In the early eighties, mainstream American comic books lagged behind some of their British counterparts which featured more sophisticated and literary dialogue and story concepts. Then, after Alan Moore reinvigorated DC's poorly selling Swamp Thing, DC editors quickly signed up other emerging British writers such as Jamie Delano ({Hellblazer), Neil Gaiman (The Sandman) and Grant Morrison (Animal Man). This proved so successful that the "British invasion" of DC continues to this day.
This was parodied in the Doctor WhoVirgin New Adventures novel Sky Pirates, which reveals Bernice Summerfield to be the author of a bizarre Vertigo-style comic called The 45 Second Piglet; said comic having been commissioned simply because she was in a big building in New York with a British accent.
Bernice was created by Paul Cornell, a British writer. Guess who he writes comics for now?
The old Chip N Dale Rescue Rangers comic book parodied this in the form of having the Rangers encounter a few other rescue groups made up of small animals (one of whom was revealed to be working for Fat Cat).
A large portion of The Dark Age of Comic Books was in some ways pretty much an extended attempt at following the leader by creating comics in the vein of The Dark Knight Returns and its contemporary, Watchmen, in an attempt to reflect the complexity and depth of these works. However, many critics — including, amongst others, Alan Moore, writer of Watchmen — accused them of only copying the superficial details, mainly represented by the Nineties Anti-Hero, rather than the storytelling complexity and experiments with medium that these works pioneered, with the result that most comics of this period were no more deep or original than the earlier works they were moving away from — they were just nastier.
For a bizarre blend of Aversion and Subversion in a package made of Reconstruction, see Alan Moore's Tom Strong, a reaction- one is tempted to call it an apology- to the darkness that followed the Watchmen comics.
Batman: Year One was the UR Example of origin stories in the more recent eras. Now both Marvel and DC produce Year One stories, with varying degrees of success, although none of them could match Batman's.
This article suggests that most of the nostalgic turn of recent superhero comic books can be tied to the popularity of Kurt Busiek's Marvels.
