Clark Kent, more so Post-Crisis (where his moderate dorkiness is natural) than pre-Crisis (when his extreme dorkiness was just a facade).
Superman's pal Jimmy Olsen. He's been the textbook example of adorkableness in comics for nearly 70 years. Artists just love drawing him with that classic sweater vest and bowtie combo. Plus he's a (sometimes naïve) Nice Guy with incredible loyalty.
The Flash: Out of all Flashes, Barry Allen comes across as the most socially awkward, clumsiest one. His lack of punctuality, terrible humor, his love for comics and anything related to science, and Wide-Eyed Idealism makes him even more adorkable. According to The Life Story of the Flash by Mark Waid, Iris once told him how sexy she found this, and "he immediately coughed an entire lemonade through his nose".
This is a trait that applies to all incarnations of the Flash, by the way. Jay Garrick's Golden Age adventures and flashbacks to it showed he could be a real dork himself in his youth. Barry's nephew and successor Wally West was the kind of character who could somehow simultaneously be the coolest guy in a room but also the biggest dork, and its his childlike dorkiness that endeared him to his wife, Linda Park. Bart Allen, moreso when he was Impulse, was an adorable little weirdo who's off-beat attitude and odd behaviour made him exceptionally endearing. Jesse Quick downplays it as she's a self-serious workaholic, but she repeatedly demonstrates Not So Above It All, has shown to have issues with anxiety, and tends to comically overreact, while her more upbeat first appearances literally cast her as the JSA's adorable bookworm niece.
Kid Devil is the single most adorkable Teen Titan, thanks to his geeky love of comics, superheroes and movies, and from usually being the underdog.
This is probably a big reason why fans love Ted Kord/Blue Beetle II so much. His successor, Jaime Reyes, shows it too, especially in Batman: The Brave and the Bold.
Dan Dreiberg/Nite Owl II of Watchmen is based on Ted Kord, and therefore also fits. It doesn't hurt that he and his Love Interest Laurie Juspeczyk/Silk Spectre II are, despite the uber-deconstructive nature of the story, the only two heroes in it that would fit in in a "normal" comic just as well. Dan's got some baggage, but it's normal, sane baggage.
Tim Drake (the third Robin) has grown up a bit, but he's always been a huge geek and when he was younger was pretty unsure of how to act around girls. But he's still seen as pretty cute, and does end up getting an awful lot of girls.
Tim Drake's adopted sister, Cassandra Cain, the second Batgirl, is incredibly socially awkward, started off as a Cute Mute, and is occasionally Sarcasm-Blind. Though she is capable of speaking now, she still stutters a lot, often uses the wrong words, and frequently quotes movies and tv shows.
Stephanie Brown, Tim Drake's long-time Love Interest, successor as Robin, and later the third Batgirl, is less of a geek than Tim but compensates by being way more of a dork. She's easily excited, constantly drops movie references, happily embraces being The Knights Who Say "Squee!", is fluent in Buffy Speak, is a huge klutz, and will make repeated movie references and quips during action. A big part of her interactions with just about anyone is her dorky charm slowly winning them over.
Bane from Secret Six is a pretty good example, at least when it comes to women. His attempts to woo a woman he has been set up with first amount to handing her flowers while stating "I wish to mate. I have brought the bargaining tokens they gave to me. Shall we begin negotiations?" Then when she compliments his eyes, he replies "...Your hips seem as though they would bear an adequate weight ratio." This is all in between the moments he's stunned speechless by her.
Static: DC hero Static is one of the dorkiest heroes around, and proudly admits that he identifies with a Shock and AwePokémon Pikachu.
Captain Marvel often has this occur with him, as an attractive woman can usually get a blush or a brief stutter out of him if she shows him some interest, and especially if she kisses him. He likewise tends to take praise from other heroes with a great deal of enthusiasm and a large smile. Just one of the ways to remind everyone that underneath the superpowers is a 12-to-15 year old kid with a big heart and a lot of compassion.
In the Justice Society of America, there's young hero Cyclone. Giant superhero and Wizard of Oz fangirl with motor mouth issues and a tendency to say the wrong thing, she squees at the littlest thing and hero-worships everyone on the team. They're all both annoyed and endeared by her behaviour.
Wonder Girl II (Cassie Sandsmark) started out as tomboyish mythology buff who made zero attempts to appear feminine but started stammering and trying to act more "girly" around guys she thought were hot outside of combat situations.
Black Knight: Dane Whitman a.k.a. the Black Knight, especially in Captain Britain and MI13. He's good-looking, skilled, and well-intentioned, but prone to cheesy heroic speeches, corny chivalrous gestures, cracking bad jokes, snorting when he laughs, sticking his foot in his mouth repeatedly, and petting kittens. His girlfriend finds the speeches and gestures endearing, and at one point when he accidentally ticks off his girlfriend's father, said father watches his bumbling attempts to apologize and notes, "It's hard to be annoyed at him. That in itself is annoying."
Ms. Marvel: The new Ms Marvel is a Pakistani-American teenager and also a massive fangirl, who writes crack fic (Avengers on Planet Unicorn, Wolverine and Storm in space (they fight a monster that farts wormholes!)), ships her mentor with Spider-Man, plays videogames, quotes memes and goes into squeeing when meeting Wolverine.
The Punisher: The Russian is an exceptionally rare evil example. He's unfailingly enthusiastic about everything, relentlessly cheerful, expresses a desire to use the money from killing The Punisher to buy "many Levis and compact disks" and is a huge fan of good superheroes, wanting autographs from the X-Men and the Fantastic Four, serving as President of the Smolensk Chapter of the "Daredevil: Man Without Fear" Fanclub, and believing that Thor, with his big hammer, would make a great Communist.
Spider-Man: Spider-Man, especially when still a teen like in the Ultimate verse. Apt to start geeking out about the science behind whatever villain that week is using, and he trips over himself in social situations a lot. Still he goes out and does the right thing because it is right.
"You'd think that young Doreen Green's mutant power was Super-Adorability, but her abilities go far beyond simple cuteness. Her semi-prehensile tail allows her to display amazing feats of agility...well, yeah, that tail IS pretty adorable. How about the empathic bond she shares with squirrels? That's a good power! But, yeah, it's also kind of adorable."
Oddly enough, while 90% of the time Cyclops is a Sugar-and-Ice Personality guy who barely ever shows any emotions, when he does he becomes one of the most ridiculously awkward people you'll ever see. As several members of the fandom have pointed out, it's hard to absolutely despise the guy when he's wearing an absolutely dorky smile or epic failing at ice skating. His younger self qualifies all the time as a Shrinking Violet.
Nightcrawler can usually be depended upon to play the role of a dashing, romantic, swashbuckler hero to any female in the vicinity... only to get hopelessly tongue-tied whenever they show even the slightest reciprocal interest.
X-23, of all people, in All-New X-Men #30. While at a club with teenage Warren, they get into a brawl with some punks who take exception to Warren's wings. During the fight one of them grabs Laura by the hair and she reflexively turns on him with her claws out. When Warren stops her with a hand on her arm, the sheepish grin she gives him◊ as she puts them away is positively adorable. Later, she and Warren have a heart-to-heart where for a brief moment Laura becomes just an awkward teenage girl wondering what a guy likes about her. Then there's her awkward smile◊ when she turns to Doctor Strange for help curing three dying clones, one of whom was currently imagining creative ways to murder him in annoyance over being judged.
Other Comics
Jeff Smax, The Big Guy in Top 10. Though he starts off as being rough and uncommunicative to his new partner Robyn and much of Precinct Ten, he gradually starts opening up, and is even visibly distraught when Robyn is nearly killed and trapped under wreckage by a rampaging perp. It really starts coming out in his own mini-series Smax, when he returns to his own dimension. There, it is revealed that much of his gruff demeanour comes from having No Social Skills whatsoever. He's also intensely embarrassed about his fantasy pastiche world, as well as his incestuous relationship with his twin sister.
Blondie (1930): Dagwood Bumstead could be easily the poster boy for this trope (as well as one of its earliest examples) — He is not only rather nerdy, but also extremely lazy and awkward. To top it all, his wardrobe is still stuck in the 1930s, bowtie and all (although he has stopped wearing hats).
The Transformers: Robots in Disguise: Thundercracker, powerful Decepticon and member of the elite Seeker team in Megatron's army, known for his sonic booms that can deafen and overpower friend and enemy alike. Post Heel–Face Turn he lives on Earth, with an adorable dog and writing truly terrible Fanfiction, er, Screenplays which features a character named Josh Boyfriend who is "handsome like an F-22 fighter."
Over in The Transformers: More than Meets the Eye, the main cute, awkward characters are bespectacled (yes, despite being a big robot) psychiatrist and model ship enthusiast Rung who is, unbeknownst even to himself, the Transformers' God, and intellectually gifted but socially awkward engineer Nautica.
Nautica: And look! Reading! I can read upside-down! I'd always wondered if I could! Skids: You never thought to, I dunno, turn a book upside-down? [Beat Panel] Nautica: (burying her face in her hands) I'm not here and you can't see me. I'm hiding behind my wall of shame.
Andy from The Internship is a clumsy and awkward geek who is a 25 year old virgin. He is hopelessly clueless when it comes to sex and he also pulls off the goofiest and silliest expressions.
In Rainbow Brite Wisp and Willow love LARPing and pretending to save the day from evil, and are determined to stay in-character even when getting sandwiches.
◊ as she puts them away is positively adorable. Later, she and Warren have a heart-to-heart where for a brief moment Laura becomes just an awkward teenage girl wondering what a guy likes about her. Then there's her awkward smile
