This trope more or less sums up Marvel Universe character Jamie Madrox's power (triggered via physical impact), especially once he learned that his duplicates were independent and developed their own personalities the longer they remained separate from him. Only he doesn't stop with "at least one evil twin". Any personality trait can be literally embodied in one of his "dupes"; as well, he sent out dupes to master certain occupations and skills, which after the dupes are reabsorbed, he learns as well. (The latter is distinctly implied to be the cause of the former—though the "independent duplicate" problem had some prior precedents, it became a permanent issue after he had reabsorbed many of the dupes sent out to learn new abilities.)
Warning: Combining the 'master certain occupations' and the 'evil twin' problem is VERY BAD!
In the Earth X continuity, Madrox is an information broker, said to have a dupe in every major city on the planet. However, under dire circumstances and threat of starvation, Madrox is forced to eat one of his dupes, and falls victim to the curse of theWendigo—which is passed to every duplicate around the world. The heroes of Earth are forced to Kill It with Fire to eradicate the threat of self-duplicating, flesh-eating monsters.
Another Marvel Universe character, Flashback, has a similar power—but he pulls his 'clones' from his own future.
One of Flashback's future selves gets killed during a battle with Alpha Flight. This causes the original to go into Villainous BSOD because he now knows HOW he's going to die but not WHEN, which causes him to become afraid of being pulled into the past (which apparently he cannot control) because that could be the time he dies.
Damian Tryp also falls into this category, even though he's only pulled two versions of himself into the past. Doesn't stop him from causing a hell of a lot of trouble for Madrox and his team.
The Superman villain Riot's powers work the same way as Madrox, with the added ability to clone himself at will.
This was shown in Marvel's Lighter and Softer Marvel Age imprint when, via some contrived means involving the aforementioned Jamie Madrox, the Incredible Hulk ended up with his power (including the physical impact trigger), filling the city with hundreds of different-personality Hulks wreaking havoc, due to the majority of the Hulks (predictably) reacting to duplicates of themselves by hitting them. There was even a hopeless romantic Hulk.
Again from the Marvel Universe: when he attempted to get his powers back via the Terrigen Mists, Quicksilver acquired this instead.
Bart Allen (formerly known as Impulse and Kid Flash) could produce avatars of himself which could travel through the timestream.
Dr Bedlam (later Baron Bedlam) from DC Comics' New Gods is a disembodied intelligence who can possess a number of robot bodies.
One The Avengers story had all the different models of Ultron sharing a consciousness. A more recent Superman story did the same thing with Brainiac, whose mind not only inhabits every version of himself through the years (except the rebellious Brainiac 2/Vril Dox), but also an entire army of robotic drones.
Doctor Manhattan exhibits this ability in Watchmen, being intimate with his girlfriend Laurie with a copy of himself while a third is continuing to work in the lab.
Invincible has the Mauler Twins, who are apparently a super-strong villain/mercenary type and his clone, neither of whom knows which is which because the original's memories were duplicated along with his body. This results in them working together relatively peacefully aside from the constant dispute, because while the idea was for the clone to serve the original, their shared memories essentially make them equals. Eventually, one of them is killed and the other has his body half burned and tries again, resulting in the original lording it over the new, obviously unburned clone until he gets poisoned and the clone tries again and manages to restore the status quo with the argument becoming about which is the lower-generation clone but once more unsolvable.
Dupli-Kate from the same series is a heroic version. Her brother Multi-Paul is out there somewhere, as well. Also, she's not as dumb as any of the guys who send all their selves off to battle. Among other things, she normally keeps her real self far away from the fighting, so that even if all of her clones are killed she'll be completely safe.
Triplicate Girl can, of course, split herself into three. Sometimes they're color-coded orange, white, and purple, (all of them are real regardless, but the cool thing is that the combined TG has a tricolor costume which splits into three mono color ones) and sometimes one of them dies while they're separated, making her into Duo Damsel. In the Threeboot continuity, the original Triplicate Girl was the sole intelligent survivor of some sort of apocalypse on her homeworld and could split and reemerge any number of selves, but when she/they was/were contacted by the United Planets, she/they split off the Legion's Triplicate Girl, who can only split out to three and won't be reabsorbed by any of her home selves because she's become too different and it worries them.
The original/Retroboot Triplicate Girl lost one body fighting Computo, becoming Duo Damsel. Then, she sent one of herself back in time to help Karate Kid with a mission in the 21st century, and this "Una" was also killed. With only one body left, however, she discovered she'd become able to make unlimited copies of herself, and took the new name Duplicate Damsel.
Minor Top 10 character Multi-Woman can make several versions of herself, each with a different superpower.
Ninja High School had Jeremy accidentally photocopy himself into 100 variations, each one with different aspects of his personality.
Animal Man possesses the power to take on the abilities of any animals in his immediate vicinity. During the Grant Morrison run, in an arc where he was trapped in a cell with no animals nearby, he took on the attributes of a single-celled organism within his digestive system. Notably the ability to replicate himself through mitosis. Within ten seconds, he was able to outnumber his opponent.
Wisp from Dreamkeepers has this as her power, splitting up parts of her personality as needed.
In the The Authority, the Engineer can create clones of herself. It got to the point where she makes herself into a literal army. There are instances in which she has sex with Jack Hawksmoor while giving a speech at United Nations, or destroying entire fleets of mooks while resuscitating a kid.
Her case is a little different, though, as she creates robots, not clones - all her other selves are still controlled by her.
Fantastic Four: Happened to Ben Grimm once, when his exoskeleton began shedding larvae that turned into clone-Grimms. The problem? They were almost allEvil Twins. Reed was forced to trap Grimm and his duplicates in an alternate dimension while he looked for a solution. Ben, meanwhile, fought off his evil versions with the help of the few good clones he could find.
As fractured as Shade, the Changing Man is on the inside, it was probably badly advised for him to split up physically. After generating an Evil Clone who escaped and attempted to supplant the original, Shade stopped deliberately duplicating himself.
In Dark Empire, Luke Skywalker is able to make one doppleganger to rescue his friends and help them escape the Big Bad while he stays behind to fight said Big Bad. The doppleganger dissolves into light soon after entering hyperspace. He never has this power again, and Leia refers to it as a Sith technique.
New Batman villain White Rabbit is revealed in Batman The Dark Knight #7 to be the duplicate of Jaina "Jai" Hudson. Jaina's pretty smart about this: her "duplicate" has different skin tone, eye color, and hair color, and wears an incredibly distracting Playboy Bunny outfit. Even then, Batman nearly figured out the truth, but was thrown off when "White Rabbit" appeared while he was on a date with Jaina.
Flash rogue Mob Rule discovered his ability to duplicate when he was dismembered by his enemies and the severed body parts grew into new people. His duplicates are completely independent from him, developing new personalities, and eventually growing to hate the original.
Legends of Zita The Spacegirl introduces Doppelganger, a small alien circus performer, who as his name suggests can make multiple copies of himself. As a bonus, each copy has a leotard with a unique symbol and/or color. (See Web Comics folder for another Zita example.)
In a Dannish Mickey Mouse story All Of Me, Mickey uses Eega Beeva's copying gun to make a clone of himself so that he could help Minnie shopping and go to a video arcade with Goofy. Unluckily, the clone starts replicating himself on its own and soon enough, there's a dozen of Mickey clones that keep replicating. When it gets out of control, the clones steal the gun and try to replicate the whole Earth for themselves... from an airplane. It only results in a copied piece of land appearing in the middle of a river as a new island, which clones decide to inhabit anyway, until there's so many of them that the island begins to sink under their weight. Eventually, Eega Beeva saves the clones by teleporting them all to a nice uninhabited planet he saw once, making sure to stop the cloning process first.
Gold Digger Villainess Array has the ability to not only create multiple bodies, but also give them different skills/abilities (or even a different species). Her name comes from the side effect of this power: when she recalls the body, it's consciousness stays behind in her head. Fortunately, they're all pretty much of the same mind (they all love her boyfriend, for example), and if she needs the same ability again, she can re-create any of them when needed, so her mind only gets more crowded if she needs some new power/skill set.
A variation with X-23, herself an Opposite-Sex Clone of Wolverine: Laura lacks the ability to generate clones of herself, but the scientist heading the project to create her intended to mass produce her and sell the additional copies to the highest bidders. Laura's mother put a stop to this by sending her to destroy the embryos during her escape from the Facility while they were still in their test tubes. Later, the demon Blackheart actually succeeded in cloning her (and even worse, bonding these clones with a symbiote). During Laura's confrontation with her "sisters" she briefly mused on whether they were sentient beings with the same right to exist as she did, but decided that regardless they were a significant threat that needed to be destroyed.