New Super Power

A standard episode concept. Our main character who has new super powers has to cope with his newfound ability, learn to control it, and come to terms with it.

Ultimately it will solve his problem that week. Half the time it's never important again, but the other half of the time it becomes a major step forward for the character. New Powers as the Plot Demands is when this is Ass Pulled.

Anime and Manga

 * El-Hazard: The Magnificent World hinges part of its overall story arc on the discovery that the visitors from Earth have each gained some special power during their transit to El-Hazard.
 * develops NEXT abilities in the second half of Tiger and Bunny.

Comic Books

 * Superboy the clone of Superman, started out with tactile telekinesis which allowed him to simulate Superman's basic Flying Brick powers and stayed this way for over a decade (real life time), but starting shortly before he joined the Teen Titans, he started picking up Superman's abilities in earnest. As of the Blackest Night series, Superboy now has all of Superman's powers having manifested freeze breath just in time to start a black lantern ring. The reason for the shift was the retconning of his orgin to make him a true hybrid clone of Superman and Lex Luthor.
 * Spider-Man has experienced this. Tying in with the live-action film series, he was granted organic webshooters. After the events in "The Other", he developed night-vision and stingers within his body. Since One More Day, however, these abilities have been Ret Conned away.
 * Years ago, several mutants developed secondary powers well after their initial abilities had become apparent. For example, Emma Frost can turn into diamond, though it inhibits her psionic talents. In recent times, Psylocke has developed telekinesis on top of her (apparently temperamental) telepathy.
 * In Buffy the Vampire Slayer Season Eight, the titular character is apparently killed in the penultimate climactic battle, only to wake up several hours later and discover that she is now a Flying Brick. By the end of the season, she is back to normal.

Fan Works

 * Sort of half of the point of With Strings Attached. The four get magic, have to learn to use it (with varying levels of success), and keep developing new stuff, especially Paul.

Literature

 * In the Wild Cards series, Roger Zelazny's character "The Sleeper" gets his name because every time he wakes up he has a different set of superpowers. He stays awake for a few weeks (stretching it out with meth), then sleeps for a few months, and does it again. After several years of this, he develops a routine upon waking. It goes something like, "I couldn't pick up my bed with one hand, so I knew it wasn't super-strength this time, and I cut myself shaving, so it couldn't be invulnerability. Exercising my throat only made it rasp, so I didn't have the subsonic vocalizations I had that one time either..."

Live-Action TV

 * Smallville features this trope in spades. When the series starts Clark only had the speed, invulnerability, and strength of latter-day Superman. Over the series each of his abilities (X-Ray vision, heat vision, super-hearing, telescopic/microscopic vision) has developed, some spontaneously, some as a result of a condition (i.e. hearing when he became blinded). Flight has already been hinted at, but it's unknown if it'll appear in the series due to the creator's "No Tights, No Flights" rule.
 * Smallville had sexual tension linked to Clark's heat vision, temporary blindness leading to super-hearing, and super-breath manifesting in a sneeze.
 * Honorable note, Clark's heat vision problems happen again when he loses his memory. Thus lost control of all his powers and had to relearn them. The same happened with his X-Ray Vsion, Super Strength, and Super Speed—which all happen in a matter of hours, as opposed to the three seasons it took the first time.
 * Other people, such as Lionel Luther, have run into the heat vision problem when possessing Clark's body and/or powers.
 * The Greatest American Hero was entirely based on this trope, as was My Secret Identity.
 * The entire premise of Heroes. Quite a bit of some incarnations of X-Men too.
 * The best example in Heroes which isn't as a result of power-collectors like Sylar, Peter and Arthur is when the living bomb guy (Ted) gets taught how do an EMP (by the non-powered HRG, who just somehow knows Ted can do this) in late Season 1.
 * Happens in My Secret Identity at the start of the second season.

Web Comics

 * Aylee goes through this several times in Sluggy Freelance due to hers species' involuntary Shapeshifting ability. Her abilities have included flight, giving off electromagnetic pulses, breathing fire, having a nearly impenetrable spell, sprouting spikes, and being over 100 feet tall. Normally acquiring a new power means losing her previous ones, though she has retained her wings through most transformations.
 * The Inexplicable Adventures of Bob plays with this when Bob briefly gains superpowers that he can't control well.

Web Original

 * In the Whateley Universe, this is one of the basic tropes for most everyone. The setting is the Super-Hero School Whateley Academy, where mutants from all over the world go after getting their Puberty Superpower. Few of them can get the best from their powers without training, and even those with the obvious powers (Lancer has the standard flying brick package) can learn better ways to use their abilities. In particular, Lancer's back story has him having to learn to use his powers in a hurry when his brother ratted him out to the local mutant-haters on the army base, leading to a fight in which Lancer had to bust up a couple jeeps and an Abrams tank to get away.
 * Tennyo is probably a better example.
 * Jade toes the line between this and "New Powers as the Plot Demands." Note that most of her 'limitations' are probably psychological in nature.
 * The Global Guardians PBEM Universe used the Hero System rules regarding Experience Points, thus allowing characters to purchase new powers. Restrictions were put into place, however, that only allowed New Super Powers that made sense when compared to the character's base concepts. Several storylines involved the player figuring out how to use these new powers.

Western Animation

 * Attempting to deal with her newfound powers in Transformers Animated caused She reduced herself to shooting small energy blasts and using her jetpack for the rest of the series and never returned to her more superpowered state--thankfully.
 * The Powerpuff Girls, "Ice Sore": Blossom develops freeze breath on the hottest day of the year. But when her use of it accidentally lets some robbers make a clean getaway (freezing the robbers' getaway car, as well as the road, actually allows it to slide to safety), she vows not to use it anymore. Right after that though, her sisters literally have to beg to get her to use her ice breath against a huge fiery meteor that's about to crash into Townsville.
 * Ben 10, "Big Tick": Ben, to his surprise, discovers that the Omnitrix can turn him into more than ten aliens, and becomes Cannonbolt. Cannonbolt doesn't breathe fire or shoot freeze rays from his eyes, and is likewise devoid of magnetic blasts. In addition, he has problems balancing. The form can, however, curl up into an armored ball and roll, alternately ricocheting off of things or smashing through them. When all of Ben's other forms fail to stop the immense, world-destroying Monster of the Week, Ben must, despite his initial misgivings, use this form to save the world. Cannonbolt is actually useful again a few episodes later against a recurring antagonist who had, in an earlier episode, absorbed the powers of Ben's original ten forms, and goes on to become one of Ben's more frequently-used alien forms.
 * Starfire of Teen Titans thinks she's becoming a monster, turns into a chrysalis, nearly gets eaten by a chrysalis-eating monster, and... comes out as her old self, but with Eye Beams.
 * Freakazoid! parodied this and hung a lampshade on it in the first episode of its second season, with Freakazoid struggling to develop telekinesis because "he needs a new power".
 * A constant plot element used in Danny Phantom. By the end of the show, he has the powerful ghostly wail, an ice element, and even managed to duplicate in four (a power he's been trying to work for months), among others.
 * In Generator Rex, Rex at one point