The British Anthology ComicThe Beano started in the 1930s but didnt really grow the beard until the 1950s when its most popular characters/comic strips began appearing such as Dennis the Menace, The Bash Street Kids, Minnie The Minx and Roger The Dodger. Although the comic did have it's highest readership before these characters were introduced but none of the characters from before the aforementioned four remain in the comic or are as popular (or long running) as these four.
The first three Asterix books are decent enough, but René Goscinny and Albert Uderzo were still clearly trying to find their feet. The fourth book, Asterix the Gladiator saw a major improvement in both artwork and writing, along with Julius Caesar becoming a much more fleshed-out and interesting villain. A slight slip-back then happened with Asterix and the Banquet, a solid enough story but one that ultimately didn't do much to advance the story or characters, and had little appeal to non-French readers due to it being a cross-country tour of France with the heroes collecting local delicacies. And then came Asterix and Cleopatra, which saw the artwork and writing both taken to the next level (along with properly introducing Dogmatix), and is still widely considered the best book in the whole series.
Cerebus the Aardvark when it started to undergo, well, Cerebus Syndrome; depending who you ask this could be as early as the mini-arc that introduces Lord Julius or as late as partway through High Society, so anywhere from issue 14 to 30-ish, with the first Mind Games issue (20) and the beginning of High Society (26) being particularly popular places to draw the line.
Doctor Who (Titan) sees this with its Twelfth Doctor title. Most of the first "year" of stories ran alongside/between Series 8 and 9's runs on television, and thus were unable to incorporate much of that continuity. While alternating writers Robbie Morrison and George Mann provide spot-on dialogue and characterizations, most of the Year One stories come off as slightly generic adventures that could easily be rewritten for other Doctor-companion teams (particularly "The Swords of Kali"); it isn't until the final story, "The Hyperion Empire", that Twelve's TV exploits are incorporated in a meaningful way. Year Two, which brings in Rachael Stott and Mariano Laclaustra to alternate the art duties, sees a huge jump in the visual quality, and the writing takes full advantage of the TV continuity by bringing in a variety of fan-favorite villains and giving more page time to established secondary characters before taking the Doctor's story beyond Series 9.
The first year's worth of comics are largely self-contained stories (with one 2-issue story), many similar to the type shown in the later cartoon series, and it doesn't appear that anything really important is happening. It wasn't until issue #14, with its introduction of Destro into the comic book, that the series really took off. Destro stirred shit up with his first appearance, taking Baroness's loyalty from Cobra Commander as well as rekindling their former off-camera love affair. And unlike the cartoon, where Destro was 100% loyal to Cobra Commander, Destro's relationship with Cobra Commander hit the bricks almost immediately when Cobra Commander tried to kill Destro to keep him from (potentially) usurping his leadership, culminating in a botched attempt on his life that nearly killed the Baroness and led to a massive multi-year war between the two men over the course of the Marvel series.
Others consider the "beard growing" moment to be issue #21, which not only introduced Storm Shadow (who became so popular with fans that Hasbro agreed to turn him into a good guy), but was Larry Hama's ambitious "Silent Interlude" issue, an experiment in doing an entire comic with no dialogue or sound effects whatsoever. The issue garnered much critical acclaim, helping GI Joe go from lame toy tie-in comic to an actual well put together comic book in the eyes of many comic fans.
The Goon was the first ever comic series by writer and artist Eric Powell, and it shows. Less than a year and a couple of publisher switches later, he is producing one of the best ongoing series' around.
Speaking of arrow gimmicks, Marvel's very own Hawkeye, real name Clint Barton, struggled with finding a niche since his introduction. Part of the problem was that, as a member of The Avengers, being a guy with uncanny aiming abilities and trick arrows is not all that impressive when fighting alongside a near-immortal super-soldier, a billionaire with access to a robot super suitand near-unlimited economic resources, and the literal god of thunder, among others. For years, writers struggled to keep Clint relevant, even having him shun the Hawkeye mantle in favor of other alter-egos, namely as Ronin (originally belonging to Maya Lopez) and Goliath (originally Hank Pym), but these changes did little to give poor Clint an identity of his own. Then Hawkeye was given a new run in 2012, written by Matt Fraction. Fraction's run finds a very clever solution to give a boost to a character with a seemingly underwhelming skill set for epic global-stakes battles: to focus on his civilian life instead (The prologue to each volume even spells it out: "This is what [Clint] does when he's not being an Avenger"). This run of Hawkeye plays less like a traditional super hero comic book and more like an indie slice-of-life comic about Clint Barton, a guy from Brooklyn, who also happens to be a super hero. Even the art style (mostly courtesy of David Aja) is different: the comics feature flat and minimalistic colors, rough brush inking, and relatively realistic and subdued body proportions, adding to that relaxed DIY indie comic feel. This run is beloved by critics, and one of its issues (#11) even won an Eisner.
It took a while for The Incredible Hulk to settle down on how they would go about their portrayal. Hulk ranged from good, to downright evil, from Banner's intelligence to being completely incapable of abstract thought. They were even unsettled in color, and how he transformed. It wasn't until the first Hulk Annual that they settled down in, green child-like Berserker, who speaks in Hulk Speak, and is powered by Unstoppable Rage.
Judge Dredd underwent this in '78-'79 (its 2nd and 3rd years). The Cursed Earth Saga helped build up the world Dredd lived in. The Day The Law Died was one of the first big stories that was done as a single narrative, and helped define the world further. Judge Death introduced a fan-favourite villain and Judge Anderson.
The beard wasn't fully grown until The Apocalypse War in 1982, which marked a significant turning point: The strip's moral ambiguity was shoved right to the forefront, and the in-story repercussions can still be felt in stories today.
Geoff Johns' Justice League (2011) run is roundly thought to have grown the beard once Forever Evil (2013) ended. While the first arc was considered a So Okay, It's Average (at best) origin for the team, the second forgettable, the third being basically a bunch of filler (and the Throne of Atlantis issues to tie-in with Johns' Aquaman), the fourth part of the very drawn out and unnecessary Trinity War crossover (which for some reason necessitated an entirely new Justice League series be created, while also dragging in Justice League Dark), and the fifth being a mediocre Cyborg story, the actual event Forever Evil was decently liked and changed the status quo of the team while adding in some wild cards. Instead of random stories, the series was now building to a grand cosmic epic, which is what Johns is known for, once Earth-3 was reintroduced in Forever Evil. Lex Luthor and Captain Cold were added to the cast as Token Evil Teammates, throwing some more emotion into the stories as they tried to reform... or didn't.
Mortadelo y Filemón was an entertaining weekly strip with the traditional few-panels-and-punchline formula, but it wasn't until their first full book, El sulfato atómico, that the now standard supporting cast appeared, allowing for deeper storylines and better interactions for the protagonist duo. Ibáñez's drawing style also evolved notably, imitating that of Hergé's Tintin.
Paperinik New Adventures was already considered a good comics for creating a Darker and Edgier universe to Donald Duck's superhero identity, but really started running with the sixth issue, "The Day Of The Cold Sun", which showed how three-dimensional the new characters could be and that there isn't necessarily a Black-and-White Morality.
Some of Rob Liefeld's characters had this happen, specifically Supreme and Youngblood. This was mainly because after Liefeld left Image and went to Awesome Comics, he handed over all the writing duties to Alan Moore. Moore promptly deconstructed all the Dark Age stuff and reconstructed everything fun and silly about the Silver Age. The reimagining of Glory, formerly a shallow Ms. Fanservice 90s bad girl has received critical praise too
The first story arc of classic graphic epic The Sandman, lasting for seven issues, seems to set the title up as just another horror title. With issue 8, "The Sound of Her Wings", the comic introduces Death (one of its most popular characters) and led to the series becoming an ensemble series, with other characters existing alongside Dream, having adventures that Dream finds himself drawn into. With the introduction of Death, the more familiar characterization of Morpheus is established, and the series overall excised its continuity from the DCU beyond a handful of references. Neil Gaiman himself considers the issue to be this for his entire career, saying that it's when he stopped imitating other writers and truly found his own voice.
Sonic the Hedgehog (Archie Comics), although based on the darker Sonic the Hedgehog (SatAM), started out as light comedic material in the vein of Adventures of Sonic the Hedgehog that constantly played fast and loose with both the fourth wall and the idea of any kind of continuity. It slowly grew more serious as it went on, adding more serious elements and having more of a continuing plot. The beard began its stubbly growth in issue 17—not so much the story itself, but the Princess Sally story that would be resolved several issues and a special later. Issue 19 featured the first full-issue epic with some real major stakes behind it. By issue 21, the overall tone had taken a rather sudden turn to the darker, and the beard was fully grown by issue 25 which pulled off a decent Compressed Adaptation of Sonic the Hedgehog CD, which of course included Metal Sonic. The comic then came to its head with the climax of the Endgame arc in Issue 50.
Similarly, Sonic the Comic grew its beard with issue 8 when a time-travel plot shifted the setting to one of Mobius ruled by Robotnik and Sonic leading freedom fighters against him.
Sonic the Hedgehog (IDW) was, for the first 15 issues, criticized for being formulaic, easy to predict, and without real stakes. Then the Metal Virus/Zombot arc started, adding high stakes, a genuinely creepy threat, and Sonic showing regret over past actions, which satisfied fans who wanted the comic to be more developed and thoughtful.
Superman (Phillip Kennedy Johnson) opened with an arc that was considered very average, just being a father-son adventure with Clark and Jon Kent punching aliens, with a lot of excessive narration. However, once the story proper kicked into high gear with the introduction of the Phaelosians and Warworld, which gave Superman opposition that he couldn't just punch away, established a dark world that put the Man of Steel through the wringer and justified the abundant world-building, and had him form his own version of the Authority and leave Earth, which also allowed for Jon to take up his mantle as Superman.
Supergirl's early stories in Action Comics since her first appearance (The Supergirl From Krypton (1959)) were self-contained tales where Kara helped orphaned kids or secretly stopped run-of-the-mill criminals. The Unknown Supergirl, written by Jerry Siegel, introduced her first nemesis, altered completely the status quo (Kara got adopted, left the orphanage and became a public hero), and marked the shift to longer storylines where the Girl of Steel fought super-villains.
Teenage Mutant Ninja Turtles Adventures started out with adaptations of the cartoon, then followed it up with a couple of short humorous story-lines. By issue 10 it had shifted to an ongoing story, but it truly hit its stride around issue 29 when Ninjara was introduced.
Tintin began as a series of rather childish wish-fulfilment adventure yarns with a cliffhanger, followed by a ridiculously improbable escape, on every other page. The maturing moment came with The Blue Lotus, when the creator started getting serious about his research and realistic portrayal of distant locations. The previous books had often used pretty dreadful stereotypes. The Blue Lotus, while still being quite iffy in its portrayal of the Japanese, gives a very sympathetic look at the Chinese rather then relying on the stereotypes Herge used for earlier books, as well as telling an engaging political satire on the Manchurian incident.
The Transformers (Marvel) was okay, occasionally outstanding, for the first several years of its existence, until writer Bob Budiansky got tired of it and started writing such gripping tales such as the human-sized Transformers who joined the pro-wrestling circuit or had romances with giant Amazon women in space. UK Transformers writer Simon Furman took over for the last two years of its run and the stories saw an immediate and significant upswing into epic, mythology-driven and long-running arc plots involving both new and old characters. Many old fans who'd gotten bored during Budiansky's reign returned to the fold and sales saw a significant upswing (passing 100,000 a month), but unfortunately that still wasn't enough to keep it afloat, and the comic was cancelled.
Transformers: More than Meets the Eye saw artist Alex Milne grow his beard. During the Dreamwave era he suffered under the horrid management of Pat Lee, being forced to follow Lee's hideous "house style" of art like many other artists. Even after the license for Transformers switched over to IDW he still found constant critique of his art, accusations of Dull Surprise, and was always teamed with terrible colorists who made his detailed line art murky and confusing. Come MTMTE and Milne decided to give his art style a huge revamp; the result is an absolutely gorgeous style that mixes realistic detail with cartoony expressiveness. Even better, he managed to get himself teamed with colorists like Josh Burcham, who used a muted yet vibrant color palette to make his drawings leap off the page. Overnight he went a symbol of everything bad about the Dreamwave comics to one of the most acclaimed Transformers artists ever.
Speaking of More Than Meets the Eye, it's widely agreed that IDW's first Transformers universe grew the beard during what has been retroactively labelled "Phase Two", which saw the end of the Autobot-Decepticon war. The stories were now focused on post-war politics, expanding the setting with new mythology and characters and stories were more diverse, both in terms of the tone and genres as well as in diversity. Even the Transformers wiki mentions this, and it's considered a "golden age" for Transformers comics.
While the original X-Force was never a critical darling, it's agreed that the few handful of issues pencilled and plotted by Rob Liefeld are the absolute worst. As an action comic, it failed thanks to bad posing, blocking, backgrounds and character designs and as a story-driven comic it just revelled in the excess of the 90s and had every character act like a walking cliche. Once Liefeld left the book, Fabian Nicieza was able to tell the stories he wanted and develop the characters properly, and while the result isn't exactly an Eisner-winner, it made the characters more popular and was a solid superhero comic — a vast improvement over the Liefeld era.
X-Men could be seen as an example of this trope. The first 66 issues were seen as a low-rate Fantastic Four knockoff which eventually got relegated to reprints after that... However, replacing most of the entire team and getting Chris Claremont to write on it led to the X-Men becoming the media franchise that we know and love.
Subverted when they tried to rejig the title the first time by putting stalwart writer Roy Thomas and superstar artist Neal Adams (who was making his Marvel debut) on it, where they reintroduced the Sentinels, brought Magneto and Xavier Back from the Dead, introduced Sauron and the Savage Land mutants and ended on a high note when all of humanity joins wills to battle an alien threat...and then the title proceeded to get cancelled right after that.
Stan Lee even once stated himself in a discussion with fellow Marvel writers Bob Harras, Fabian Nicieza, and Scott Lobdell that he felt the same way about the comic when they took it up and improved it beyond its original format and thanked them publicly for doing so.
Wolverine when he joined the X-Men was a one-dimensional Jerkass with a disrespect for any authority, and Claremont and Cockrum considered dropping from the team. Cockrum's successor, Canadian-born John Byrne, wanted to keep the only Canadian character on the team and helped develop him into Marvel's most popular character.
The first two arcs of Y: The Last Man were a bit light on the drama, which was a tad jarring considering how much could've been done with the premise. Things got interesting in the astronaut arc, which skyrocketed the intensity in a refreshingly unexpected way.
Yorick, like Riker and Blackadder before him, has literally grown a beard by the start of that arc, although he shaves it off halfway through. Given how many other sci-fi references that series had, this may have been intentional.