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Archie Comics' Sonic the Hedgehog, although based on the darker Sonic Sat AM, started out as light comedy in the vein of Adventures of Sonic the Hedgehog. 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 the Metal Madness arc.
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.
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.
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.
The first year's worth of the Marvel G.I. Joe comic 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. Much like Marcia Cross in Melrose Place, 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.
It should be noted that Larry Hama would draw upon supporting characters and concepts introduced in several of the early issues later on, starting with reintroducing Kwinn in the Sierra Gordo arc that begins in issue #12. Issues 3, 5, and 8 are very much like cartoon episodes, while issues 4 and 9 are self-contained stories similar to the later Special Missions 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.
The entire company of Image has arguably undergone this. Image was founded by several artists from Marvel Comics, all of which were known for Darker and Edgier series, and the original titles such as Youngblood and Spawn reflect this. After the departure of founding member Rob Liefeld in 1996, things began to look up, with several lighter titles such as Bone and less Dark Age grit (which means that the early comics haven't aged well).
Speaking of Spawn it grew a beard once it became a dark horror comic.
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.
This coincided with a Growing the Beard moment for 2000AD in general: the merge with Star Lord and Tornado brought in classics such as Strontium Dog, and other fan-favourites like the ABC Warriors and Fiends of the Eastern Front started.
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.
Knights of the Dinner Table was initially a short, fairly shallow series of one-off jokes featuring Flat Characters, no ongoing story, and crude art. Over the many years of it's publication, it's evolved to a lengthy, deep series featuring fully-fleshed out characters, long story arcs, and...slightly less crude art. The turning point was probably when it stopped focusing on every strip being a brief joke ending in a punchline, and started truly dwelling on the narrative of the campaign, as well as the lives of the denizens of Muncie.
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, Valor y al toro, 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.
The first issue of Red Hood and the Outlaws was highly controversial and caused many fans to judge the series as unlikeable (Jason), shallow and sex obsessed (Starfire), amongst other terms. The book has since subverted many of these preconceptions, despite the fact that there are still some people who refuse to look past the first issue.
Issue 6, which recounts how Starfire and Jason met, is also considered pretty good.
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.
TMNT 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 Growing the Beard moment came with The Blue Lotus, when the creator started getting serious about his research and realistic portrayal of distant locations.
The US Transformers comic was okay, occasionally outstanding, for the first several years of its existence, until writer Bob Budiansky 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.
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 Jerk Ass 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.
Even 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.
Image Comics itself became an example sometime around Invincible. For much of its prior history, it was known for style-over-substance X-Men and Batman knockoffs that came out late and looked hideous. Invincible was largely a Reconstruction in comparison, and completely counter to both the traditional Image product and most superhero stories of the time. By the time The Walking Dead came out, Image had shifted its focus off trying to beat the Big Two and instead geared itself towards becoming the king of the indie titles. Today, Image is better known for books like Saga, Elephantmen, or Chew than any of its 90s offerings, which have mainly become The Artifact.