The Captain Underpants books frequently feature comics based on the title character created by George Beard and Harold Hutchins. Almost every book begins with George and Harold presenting a comic providing exposition on the series up to that point. The Super Diaper Baby spinoff books have the entire books in the same comic format.
Watchmen includes excerpts from the autobiography of one of the characters, as well as interviews with various others. Watchmen also includes the meta-comic Tales of the Black Freighter which is drawn by an artist who is missing throughout the story.
Bolivar Trask's sci-fi pulp series The Sentinels in X-Men Noir. For bonus points, the original series featured chapters from The Sentinels as back-ups.
Punisher Noir has Frank Castelione, Jr.'s favorite radio drama, The Punisher.
Iron Man Noir has Marvels: A Magazine of Men's Adventure, a pulp magazine featuring the (heavily fictionalized) exploits of Tony Stark as written by his friend Virgil Munsey and, later, Pepper Potts.
'Mazing Man's friend Denton Fixx wrote comic books. His Zoot Sputnik stories appear in a few issues of 'Mazing Man.
Spider-Man's former wife Mary Jane was an actress in a soap opera called Secret Hospital for a while. (Her character's name was "Sybil Shane" and from what we saw of the show - which was very little - her character seemed to be a vixen of sorts, and the show pretty much had every soap opera stereotype included.)
In one issue of The Sandman, Shakespeare and his actors perform A Midsummer Night's Dream for Oberon, Titania, and numerous members of the fairy realm. Since, as noted above, A Midsummer Night's Dream already contains a Show Within a Show, this makes the play about Pyramus and Thisbe a show within a show within a show.
In Supreme, Ethan Crane and Diana Dane are the creative team on Omniman, a fictional character who is even more like Superman than Supreme is. The same company also publishes Warrior Woman.
Examples of type 2 (characters are fans)
In the comic book Young Justice, the characters watched a TV show called Wendy the Werewolf Stalker, a parody of Buffy the Vampire Slayer. This eventually became a Type 3 for a couple of issues when Cissie King-Jones (Arrowette) guest-starred in an episode after becoming famous during the Sydney Summer Games.
In the Super Mario Bros. comic books, Mario is a huge fan of comic-book-within-a-comic-book Dirk Drain-Head, which is hated by the other good guys (including Luigi, who ironically looks exactly like Dirk), but loved also by Bowser's minions.
One issue of Hack/Slash has Cassie and Vlad battling a slasher at a comic book convention; needless to say, there are a few comics within the comic. The most significant one, Wunderkind, is a blatant Captain Ersatz of Captain Marvel.
Al Capp's classic comic strip Li'l Abner had the comic-strip-within-a-comic-strip ''Fearless Fosdick'', which was a parody of Dick Tracy that became almost as popular as Li'l Abner itself. Later Capp did a similar parody of Peanuts called Pee Wee.
Justice Girl is a comic within a comic in The Maze Agency (and, in universe, spawned a short-lived TV series). jen was a huge fan of Justice Girl when she was younger.
There's a running gag in Hawkeye vol. 4 #6 concerning a show called "Dog Cops", which is apparently a very popular program among members of the Avengers. We know nothing about it except that one of the characters is "Sergeant Whiskers", but its existence has occasioned manic speculation among fans to the point where fan art exists.
In Paul Chadwick's Concrete, the title character enjoys watching "Sky of Heads", an in-universe TV show about an afterlife where the heads of the deceased float around aimlessly and tell each other stories about their former lives. At one point, Concrete wonders: "If I showed up there, would I have this head, or my old human one?"
A few issues of Green Lantern in the Kyle Rayner days mention that Kyle used to be a fan of a comicbook character called the Cannoneer. One splash page shows that the Cannoneer is a cross between Cable and Shatterstar, drawn in a parody of Rob Liefeld's style. He used to be a member of a group called the Y-Contingent, who are presumably equivalent to X-Force.
The anthropomorphic comic Rocketship Rodents (itself a parody of Buck Rogers) has its own Doctor Who Show Within A Show parody called Professor Chronofur... And as it's an anthropomorphic comic, you probably know where it leads.
She also writes fanfiction for Magical Pony Adventures and Super Heroes (which in the Marvelverse would mean Real Person Fic), verging into type 1.
The 80s (fake) space opera porn film Captain Hammer Meets the Space Vixens shows up twice in Lesbian Zombies From Outer Space: the first couple pages of Issue #1 come directly from the show within the show, until we pull out and find Ace watching porn in the back room of a video store; and later in Issue #4, Ace finds Mr. Hagerty watching the same porn video in the auditorium of the local high school.
Examples of type 3 (SWAS is plot point)
A show-within-a-comic plays a pivotal role in Ronin.
There are several in the comic Y: The Last Man. The Last Man is a play written and performed by the Fish & Bicycles acting troupe (Yorick, the real last man, is not happy to discover that the play ends with him dying). The same people are seen several years later (unsuccessfully) trying to make an action movie about the radical man-hating Daughters of the Amazon, then finally end up creating a successful comic series about the last woman on Earth (Yorick is equally unimpressed with it). And when the protagonists are in Japan they watch traditional Noh theatre featuring a demon called Hitogoroshi (Manslaughter).
In the Marvel Universe, there's an actual Marvel Comics company that produces licensed comics based on the real-life adventures of the heroes. This started as early as Fantastic Four #10, January 1963. The She-Hulk series uses these in-universe comics in the title character's legal cases. DC Comics, after abandoning Earth-Prime, took this idea into their own canon.
Amusingly, since in most cases the superheroes themselves gain licensing money and are actually somewhat involved in the comic's production, it's implied that the in-universe Marvel comics are slightly more skewered to portray the heroes in a better light than our real-world versions of the same comics. The heroes themselves usually answer the fanmail in the comics, too, which leads to some really odd things being said — like Reed Richards wanting to get rid of fashion and force everyone in the world to wear a Fantastic Four-style uniform.
At one point, the Marvel Universe Marvel Comics company hired a new artist for their Captain America comic... named Steve Rogers.
While some heroes, like the aforementioned She-Hulk and Fantastic Four, are public figures in the Marvel Universe, others, like Spider-Man or Daredevil, aren't about to spill their secret identities on newsstands, so their comics-within-a-comic are only accurate as far as the superheroics go, and make up the heroes' personal lives and origin stories out of whole cloth.
Marvel once printed a series of one-shots, called 'Marvels Comics' which were supposed to be the comics that exist in the 616 universe.
Watchmen also has Tales of the Black Freighter, a dark pirate comic (since superhero comics didn't catch on in a world with real superheroes, pirate comics became common instead) which is used as a metaphor for various parts of the story and the characters' plights.
Daniel Clowes' comic David Boring has the protagonist find "The Yellow Streak," a one-shot comic by his father that seems to suggest why his parents divorced, while individual panels are used in the main story to suggest David's reactions.
In issue #17 of Matt Fraction's Hawkeye, the main character watches a TV special, which parallels his life, featuring the "Winter Friends," a group of super heroic animals (an analogue to the Avengers), including a non-powered dog who wants to take matters into his own hands (analogue to Hawkeye), who has a fellow canine friend who supports him (analogue to Kate Bishop), as well as three super-powered allies (analogues to Mockingbird, Spider-Man and Black Widow), and a group of tracksuit dogs (analogue to the "Tracksuit Draculas").
Played with in The Multiversity; various comics that superheroes in alternate realities are fans of reflect the events of the story ... because they're recording actual events in other universes that are affected.
The idea has a long history with DC, dating back to the Silver Age, where the comics of the main DC universe reflected what was going on in Earth-Two, where DC's Golden Age characters lived. (However, for the period when there was no DC multiverse, they went with Marvel's solution, as mentioned above.)