In a lot of fairy tales, there is one hero and one heroine (though The Prince and the Pauper and the like are notable exceptions), there will be a lot of emphasis on how infinitely talented the girl is, but not as much on the guy. The girl (who is usually either a princess or a regular girl who will become a princess later in the story) can have any range of talents, including singing (obviously), dancing, cooking, painting, or even kicking people's asses. This can occur in one of three situations:
She spends most of her time living in her castle/house, and likes to keep a few things in practice so she doesn't get bored. Same if she's been living in a tower. Or maybe, in some cases, it's part of her job.
She is required to be talented in order to marry the prince.
Her talents were given to her at birth by a magical being.
As for the guy? The guy may or may not be a prince, but he is usually given one of three talents:
In Shrek, we begin thinking Fiona is a typical Distressed Damsel until she gets kidnapped by some singing Frenchmen who want to kill Shrek, at which point she resorts to kicking their butts. It's revealed that since she lived alone in a tower for years, she put a few things into practice. Shrek, on the other hand? Ah, he's just a rather short-tempered ogre who doesn't socialise much and is a bit set in his ways, and who was dragooned into this quest very much against his will anyway. He gets better.
Inverted in The Thief and the Cobbler. The Cobbler has his own trade, but Princess Yum-Yum laments the fact that her father won't let her help out with anything.
The Princess and the Frog inverts the trope twice, giving us Talented Girl Regular Prince. Prince Naveen is apparently unable to take care of himself, whereas Tiana is tough, hardworking, and a great cook. This is debatable, though, since he does play the ukelele and speak French. The implication is that Naveen is a smart guy and can pick up on stuff, but his lifetime of hedonism made him unwilling to exert the effort to do so. Tiana, meanwhile, is forced to have a strong work ethic to get through her lower-class life.
Disney's Tangled has Rapunzel and Flynn. While the bulk of what Flynn can do involves stealing from people and making snarky comments (though, according to some fans, his other talent is the smolder), Rapunzel can paint, dance, cook, play guitar, sculpt, and knit, among other things. Plus her hair alone has many different uses, and she sometimes utilizes it in defending herself.
In Cinderella, the heroine cooks, cleans, sews (and tiny little mice-sized clothes, no less!), dances, while the prince won't even go after her himself (though probably justified as the handsome local prince visiting every single girl would probably cause more trouble than it's worth).
Live Action TV
The Faerie Tale Theatre rendition of Sleeping Beauty puts a lot of emphasis on this trope. Sleeping Beauty grows up to be beautiful, talented, and all-around perfect Bernadette Peters, while her prince is less than appealing. She uses her talents to "audition" for her soon-to-be father-in-law, who enjoys her performance and says she can marry his son. That being said, she would have been safe from the curse if she hadn't looked out her window and noticed how humiliatingly selfish and unimpressive the prince was, because she ran away and hid from him, and that's where she found the fatal spinning wheel. So in a sense, it is this trope that killed her.
Played with in the Enchanted Forest Chronicles. Cimorene is quite talented, what with all the unprincesslike things she learned behind her parents' back and then the skills she learned in the course of being a dragon's princess and Queen of the Enchanted Forest. In "Dealing With Dragons", this trope would be played straight if she and Therandil had actually been a couple or even a team; Therandil is dull and doesn't seem to have any real strengths to speak of. In "Searching For Dragons" and beyond, it's averted, since Mendanbar is talented in his own right.
Inverted in The Princess Diaries. Aside from being a Princess, Mia is a completely normal girl. Her main Love Interest and boyfriend, Michael, is a genius, great with computers and even managed to build a robotic arm that revolutionized medical science. Her other various love interests/boyfriends could qualify too: JP is talented, smart, and manages to write a play by his senior year that gets optioned to be a movie while Kenny is a science whiz who manages to build a bomb all by himself their junior year.
Winnifred and Dauntless from Once Upon a Mattress fit this trope to a T. Dauntless is a timid Mama's boy who hasn't even had The Talk yet, while Winnifred (a.k.a. Fred) can swim, sing, dance, wrestle, lift weights, and even survive through large amounts of wine.
Yggdra Union: Played straight as anything with Yggdra and Milanor. Not only are Milanor's only talents fighting, snarking, and stealing, but he's only around to be a surrogate for the male players; Yggdra carries the plot completely. At least they're not actually a couple.
Played with in the various The Legend of Zelda games. Because Link is the character that the player controls, he starts out able to do very little besides walk around and, occasionally, jump ledges and lift rocks. By the end of the game though, he can wield a wide variety of weapons perfectly. Zelda's skill level, on the other hand, depends on the game. In the earlier games, she was pretty much a Distressed Damsel. Later games, on the other hand, gave her psychic powers and the gift of prophecy, which she would have at the beginning of the game, usually to contact Link. Notably, Wind Waker has her starting off as a worldly pirate who laughs at the idea of Link being a hero. Some games also had her gain abilities as well, like the ability to turn into a ninja, wield magic, use a bow and arrow set, ride a horse, swordfight, and possess a suit of armor to fight.
Square Soft likes this trope, although few characters ever stay "a regular guy" for the whole game. Let's count 'em:
Final Fantasy V: Reina, Faris and Krile - a white mage princess, a pirate captain princess and a Bratty Half-Pint alien princess, traveling with Bartz... a hobo on a chocobo. He turns out to be an alien too.
Final Fantasy IX: Princess Garnet Til Alexandros IX, a white mage princess and summoner, traveling with Zidane, a regular teenage thief. ... He turns out to be an alien. She trained to be able to escape her castle, and is determined to fix things diplomatically on her own, but for the most part Garnet just makes things worse by being in the wrong place at the wrong time. Zidane ends up cleaning after her messes every time. Garnet has talents, she just really sucks at using them.
Final Fantasy X: Lady Yuna, daughter of High Summoner Braska, traveling with Tidus, a not-too-bright jock. He turns out to be a hallucination dream of the Fayth based on Shuyin. They're both equally instrumental to taking down Sin: her summons must be sacrificed and Tidus can distract Zombie!Jecht.
Final Fantasy XII: Princess Ashelia B'nargin Dalmasca, a diplomatic mastermind and leader of La Résistance, traveling with Vaan, who... is able to hold a sword.
Bahamut Lagoon: Princess Yoyo of Kahna, an all-powerful summoner, traveling with Byuu, a knight. In Byuu's defense, he's good at what he does — but he's got a supporting role at best.