Yu-Gi-Oh!: The protagonist has an ever-growing team of friends to support him, and not do much else. The Rival, on the other hand, doesn't have any, doesn't want any, and consistently fluctuates between bona fide antagonist and intolerable rich snob.
Everyone is a loner to some degree, and it does none of them the least bit of good.
The three pilots are the most lonely: Shinji and Asuka have a very hard time socializing or making friends due to losing their mothers and being abandoned by their parents when they were four, and at the end are the only people left on Earth; Rei because she has been raised in isolation and is the soul of an ancient alien goddess locked inside the teenager cloned body of Shinji's mother.
Rebuild of Evangelion has an aversion in the form of Mari, who outright states that she doesn't like involving others in her plans, deliberately cuts off the intercom when she hijacks EVA 02, and is hardly shown talking with anyone but the protagonist. She still manages to be a very positive, badass character who teaches said protagonist an important lesson.
Ash's Greninja was quite a loner back at the Froakie breeding farm, and he was rejected by the rest as a result. Froakie wasn't necesarily opposed to being friends with the rest, but simply preferred to train in solace. Of course, getting rejected by all the others resulted in Froakie becoming much more disconnected. His interactions with Ash and Ash's team result in him gradually getting more social and friendly.
On Digimon Adventure, the evil Puppetmon was defeated specifically because he had no friends. In a sense, all of the evil Digimon were destroyed because they were loners. Puppetmon was simply the only one who had it spelled out to him, by both the Digidestined and Cherrymon (his childlike nature also meant that he angrily rejected the notion that no one was his friend).
The relationship between Naruto and Sasuke reeks of this. Sasuke, a loner due to a strong, all-encompassing desire for revenge, and power for the sake of that revenge, ends up becoming a Rival Turned Evil due to his pride being damaged by both Naruto's ever increasing power and an ass-kicking/Mind Rape by his Aloof Big Brother Itachi. Though after the time skip Sasuke seems to be clearly one of the most powerful ninja on the planet.
But he's also increasingly off his nut and evil. This trope in relation to Sasuke was made pretty clear with him sacrificing Karin without a second thought to get to Danzo.
Gaara exemplified this before his defeat by Naruto. Constantly ostracized by his village, he had retreated into solitude and gone gradually more insane as doing so only strengthened Shukaku's hold on his mind.
Naruto's response to his backstory also fits. His first reaction was to fear Gaara because he thought someone who could willingly accept such loneliness must be incredibly strong.
Exception: Kino of Kino's Journey is remarkably well adjusted and prefers to travel alone with just a talking motorcycle (yes) for company. Kino doesn't hate people, but doesn't want to form attachments that would hinder the ideal of the Traveler. It helps that most people and societies they encounter are really screwed up.
The main message of One Piece boils down to "if you don't have any friends, you suck!" It's actually difficult to stay a loner in One Piece unless you're really a douchebag and therefore, deserving of it. Consider Tony Chopper, a talking, shapeshifting reindeer, who was outcast for his weirdness until Luffy's crew comes along and openly accept him.
Also, Zoro, Robin and Brooke. Zoro was even alone by choice. Also worth mentioning is Arlong, who cares as much for his True Companions as Luffy, and is a vicious bastard.
Played straight in a very realistic way by Brooke who's fifty years of complete isolation drove him mad.
Death Note does this in different ways, with L being the most straight-up example of this trope. Light seems to be his polar opposite, being very popular at school (apparently), and having a very devoted following for his alter ego, Kira. However, Light doesn't seem to be close to anyone in his life, and sees his girlfriend Misa as more of a tool to reach his goals. It turns out he and L are really Not So Different.
The Diclonius in Elfen Lied are rejected by society. As a result, most are either depressed or (more commonly) sociopathic, sadistic killers. It's worth noting that all of them are psychotic to varying degrees and/or have split personalities.
In Berserk, the half-mad main character, Guts, finally relents and allows others to follow him after wandering alone battling demons for years. They eventually become True Companions, but he goes off to being a loner again, which is one of the main causes of everything going into shambles since Griffith is obsessed with him. The sad thing is, he only left the Hawks because he wanted to be a better friend and peer to Griffith, and he believed that he couldn't do that unless he could find a dream of his own.
Princess Tutu has Autor, a character who is first introduced by several cameos of him sitting alone in a library, yelling at others to be quiet while employing Scary Shiny Glasses. Once he becomes integrated into the plot, it appears that he doesn't have many friends because he's consumed by his obsession with Drosselmeyer (as well as a belief that he's better than everyone)—although there's occasional hints that he's bothered by his position, including him having an angry reaction to Uzura calling him "Weird Autor". In the end, he helps give Rue an epiphany using The Power of Love, and also saves Fakir, possibly hinting that he's come to accept the boy as a friend.
Johan of Monster would be a aversion of this trope, as he's very charismatic... if he didn't kill everyone that ended up being his friend. Though it could be argue that he never considered anyone his "friend." The closest things he has to friends are Tenma and his own sister Nina, the people whose lives he made hell and whom he tries to mindrape into killing him over the course of the series.
While the heroes of JoJo's Bizarre Adventure slowly gather friends over the course of their bizarre adventures, the Big Bad of the series, Dio Brando, also has an ever-expanding group of friends/minions who are extremely loyal to him. It's even noted by Joseph that this is part of the reason why he's so dangerous (besides the vampire powers and time-stopping). A more direct example is Part 4 Big Bad Yoshikage Kira.
Sousuke Sagara from Full Metal Panic!specifically tries not to get too attached to people, mainly because he wants to maintain his business-like, cold way of following orders. Unfortunately for him, people just seem to be so attracted to him that, even if he doesn't want to, he constantly ends up with groups of True Companions. Gauron becomes rather angry when he finds out that Sousuke is constantly surrounded by friends, and breaking speeches Sousuke about how being a loner is a good thing that makes him strong and unique. Funniest part about it is that Gauron himself kept two very loyal girls by his side, making it more likely that his grand speech had more shallow reasons.
The plot of Welcome to the N.H.K. is how the protagonist Satou tries to get over his social anxiety and connect with people after he realizes how unhappy he is as a loner.
Averted in Mahou Sensei Negima!, oddly enough. The hero himself is a very kind person, but is also formal and uses Keigo with almost everyone. He tends to be rather distant otherwise. As it's an aversion and not a subversion, he's a loner but has no pathological case of avoiding people, he just doesn't socialize.
He also doesn't have a lot of time to socialize because he's constantly going through Training from Hell. However, he does tend to grow closer to the students who are training alongside him.
He probably uses Keigo because Japanese is not his first language, and the more formal ways of speaking are generally taught first when someone is trying to learn. Then again, he knows how to speak in an informal manner and does it with the Kotaro, Takahata-sensei and Anya, as pointed by one of the students.
It also doesn't help that Negi has a huge helping of survivor's guilt after what happened to his village and wants to keep people away, in case they suffer the same fate as everyone else.
Being alone is a common factor for 'mad' characters in Soul Eater, right up to the original Kishin who rejected everything and everyone out of fear. Whilst being alone has not been explicitly stated as being bad, the way the alternative is presented makes the implication clear.
Crona was at the point of becoming a Kishin, did not know how to "deal with" people and their soul was represented as a desolate landscape until Maka intervened. Once Medusa gets her child back, however, things get worse and Crona has recently rejected any reminder of the friends they found at Shibusen.
Black Star became more detached from people when pursuing a misguided attempt at becoming stronger. He challenged teams without his Weapon partner, becoming increasingly...odd to the point that a comparison was made to his kishin fahter. Tellingly, he got over it by listening to and cooperating with Tsubaki.
Being a lone Weapon is one of the fandom theories regarding Justin Law's Face Heel Turn, especially considering he appears to have encountered the Clown whom Maka and Soul 'defeated' through epic team-work. In fact, teamwork forms the basis for all of the significant victories of that pair (if not all of the meister/weapon groups) in the manga.
While Stein does get on with people to an extent, he does not exactly seek out company. Marie leaves Death City with Stein because "he is always alone". Spirit was told by Shinigami to keep an eye on Stein, something which he has apparently been doing since childhood.
There is, however, one arguable aversion with the wizard Eibon. In both the manga and anime, he's spent centuries alone either living in solitude on Lost Island or sealed inside a demon tool, yet seems no worse for wear because of it (nor is his solitude remarked upon). He even provides assistance to the heroes in story, before (at least, in the manga) choosing to remain a hermit.
Shizuka of Bakuman。 is a basement-dweller who writes truly messed-up plotlines, barely acknowledges his beleaguered, well-meaning editor, and regularly sports truly terrifying facial expressions. When this boy snaps, it'll be something on the order of the Staff of the Magi.
Yukiteru Amano of Future Diary is very much a loner at school, due to his own reclusive and anti-social personality. The other kids consider him to be a freak. He does grow out of it later on and gains a small circle of friends after deciding he never liked being alone, but unfortunately for him, Yuno Gasai considers herself the only friend he needs.
Konata from Lucky Star, while not being a true loner, often gets berated for her love of solitary activities, like playing video games or watching anime shows.
Kagami in particular sees Konata as the stereotypical otaku and/or hikikomori, and considers Konata as a potential criminal. (To be fair, Konata lacks most stereotypical otaku and hikikomori traits.)
Tenshi/Kanade Tachibana of Angel Beats! is placed in this category, but is rather a Boo Radley. She's seen as a freak because she doesn't have any friends, but that's only because She's actively trying to graduate those around her, and thus is never really seen with anyone for any length of time.
Averted in Amanchu!. We never see any of Hikari's friends from before she met Futaba and the rest of the diving club, but she is shown to be cheery and outgoing, although perhaps a bit eccentric.
All over the place in Puella Magi Madoka Magica. Homura doesn't have any friends because she was hospitalized for a long time, Mami doesn't have any friends because she must distance herself from others due to her job, Kyouko doesn't give a damn about humans after all she went through, and Sayaka actually become freaks, in so many ways, when she decides to be a loner. In contrast, Madoka try to connect with them all.
In Saint Beast, Shiva doesn't want to get close to anyone except Judas and also has a habit of putting down others. He gets treated with suspicion by other angels for his lack of community values.
In Heaven's Lost Property, Eishirou Sugata is supposedly this, complete with living in a tent, by a river, on a FRIKIN' mountain. And yes, he is weird as hell, according to Tomoki Sakurai, who is weirder, so it's weird coming from him. Though, after meeting Tomoki and co., he kinda started following them around.
Played with in Fairy Tail. Marakov finds Gajeel bumming around after Phantom Lord was defeated. When Gajeel insists that he is fine by himself, Marakov points out that some people prefer solitude, but no one likes being alone.
Zig-zagged in Haruhi Suzumiya with the title character. She starts out a loner—no friends, refuses to join any clubs—and is seen as a disruptive weirdo by pretty much everyone. Then she starts the SOS Brigade...and becomes ten times as destructively obnoxious now that she has a team of "friends" to do her bidding. But, after a while, she experiences Character Development and becomes less of a jerk thanks to being around other people; although they're trying to pacify her more for the sake of the stability of the universe than for her own benefit.
Shiina in Gourmet Girl Graffiti is a bit standoffish and does things on her own pace. While her mother is understandably concerned about her lack of friends and her being a social misfit, there're no evidence of her being seen as a freak, and she has two friends.