Randomly-Gifted

Characters who are Randomly Gifted have developed Magic and Powers pretty much out of the blue. Specifically, even the children of Muggles have a chance of randomly developing powers without the need for any kind of Superpowerful Genetics or other power granting trope having to come into play. Usually this is a setting-wide feature, allowing anyone anywhere to spontaneously develop abilities, and as such it can be used as a form of time delayed Mass Super-Empowering Event.

These characters may develop their powers immediately at birth, adolescence or later in life, and there may or may not be means of detecting these powers before/after birth. The children of Randomly Gifted characters face an interesting situation; in some settings the randomness is such that these children are no more likely to develop abilities and may become a Muggle Born of Mages, while in others they have a much higher odds of getting powers.

Society at large may be completely fine with this, see it as an honor to have a family member with special gifts, or consider it blasphemous and dangerous. Though these characters are not necessarily a new race or Witch Species they will usually be called some form of Differently-Powered Individual; saddly it can devolve into Fantastic Racism or Muggle Power and lead to calling these powered characters witches, Mutants, or demons.

Finally, the reason for children being Randomly Gifted can vary, it could be a spiritual agent, very complex genetics, or a never explained source.

Anime and Manga

 * In Tiger and Bunny heredity seems to have little or nothing to do with being a NEXT.

Comic Books

 * The X-Men and mutants are born randomly among the population, though active mutants are almost always guaranteed to have mutant children regardless of if their partner is a human.
 * In a notable exception, Graydon Creed is the son of Sabretooth and Mystique, who are both mutants.... and he's completely human.
 * In Invincible it's not known for sure whether Mark, the Half-Human Hybrid son of Omni-Man, will ever develop powers...until he does.

Film

 * This is true in Star Wars: Force-sensitive children can be born to Muggle parents.

Literature

 * In the Harry Potter universe magic-wielding seems to be somewhat heriditary, but it's possible for a wizard/witch to come from two Muggle parents, or for a non-magical "squib" to come from a Wizard family. What's interesting, and mostly ignored by "Pureblood" families, is that muggleborns tend to have more talent, and half-bloods such as tend to have more power.
 * Word of God is that Muggle-borns have a witch or wizard somewhere further back in their family tree, to provide a source for the trait.
 * This is how Talents are handed out in Necroscope you are just randomly born with powers (admittedly people whose parent's had a Talent are more likely to get one, but like the X-men Talents mostly come from random members of the population born with one).
 * Due to Background Magic Field of Echo, everybody in the eponymous city in Labyrinths of Echo can use magic. But the exceptionally talented are born seemingly at random, although living in Echo increases the chance to inherit magical skill.
 * Belonging to the Others is random in Sergey Luk'yanenko's Watch universe. Children of Others are almost always regular humans. Ensuring the birth of a Light Other from a carefully chosen couple of Light Others is a generations-long plot run by the Night Watch.
 * Iar Elterrus:
 * The Gifts, including magic, are mostly random among all species in the Aarn universe.
 * Magic is a random trait in the Nine Swords multiverse. This is little-known and a mage looking for gifted individuals has to be told to scan his city's slaves.
 * Magic runs at random for humans in Valentin Ivashchenko's settings, although the benevolent kingdom / empire tends to accumulate mages in the nobility. In Warrior and Mage, the crown prince being a mage is noted as exceptional.
 * Magic runs at random in Igor' Dravin's Alien / Xenos (Чужак) series, which can take a turn for the worse should an unsuspecting child in an overly religious area suddenly demonstrates necromantic abilities.
 * Vadim Panov's Secret City: this trope distinguishes humans from other species. Humans can be born as regular mages or Healers, and the amount of mages is increasing because of the millenia-long exposure to the Secret City's magic.
 * Vitaliy Zykov:
 * Way Home: this trope is in effect for the human factions in the setting, including the protagonists kidnapped from earth - Oleg becomes an Earth mage, while Yaroslav combines multiple theoretically mutually exclusive magic systems.
 * Conclave of Immortals: the Transition event randomly affected humans. The common powers are shapeshifting and dreamwalking. The rarer taming power seems to combine aspects of both. Other human factions within the survivors gain cleric-like powers, but this may be a case of magical Lost Technology or resurfacing religions, as some weird cults gain power after the Transition.
 * This trope is in effect for humans in most contemporary russian fantasy books, sometimes used in contrast to whatever Witch Species the author mentions as well.

Video Games

 * Bloody Roar: This is the reason Tylon and other organizations conduct experiments on zoanthropes, behind the scenes, as there are those who are born with the ability to transform into fighting beasts. While no one knows the reason for it, it was determined that it doesn't stem from genetics. Long, for example, was born with the ability while his sister was not.

Web Comics

 * Girl Genius has The Spark of Genius. It can be inherited, but it is implied to be inconsistent: children of dynastic Sparks are carefully vetted by their parents to ensure that they inherit abilities equal to or greater than their own(with the implication that they'll be disowned or dissected should they fail to impress), but it's not unknown for Sparks to be born to normal parents. These unlucky bastards have to overcome numerous obstacles to reproduce(insufficient education to make use of their talents, insufficient talent to make use of, insufficient luck to not be killed by their first inventions, insufficient solitude to not be killed by angry mobs), so simple selection pressure tends to enforce rarity.

Western Animation

 * In Avatar: The Last Airbender people are born with the ability to bend pretty much at random, although bending ability does sometimes follow family lines. One notable instance of the curious nature of Bending is an episode with twins: one of whom can bend and the other cannot.
 * Spirituality has something to do with it as well: as the most spiritual nation, the Air Nomads were 100% Airbenders.