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.
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 sweatervest and bowtie combo. Plus he's a (sometimes naďve) Nice Guy with incredible loyalty.
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.
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's (aka Batman's sidekick Robin) 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.
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.
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.
Dane Whitman a.k.a. the Black Knight, especially in Captain Britain and MI-13. 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."
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.
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.
Punisher villain 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.
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."
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!)), plays videogames, quotes memes and goes into squeeing when meeting Wolverine.
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.