One Person, One Power

A fairly common subtype of settings with a Mass Super-Empowering Event in the backstory. Each superpowered person gets one power. Some powers are more versatile than others, of course. The only exceptions are people whose one superpower is the power to copy or steal powers. Nobody gets Combo-Platter Powers as a default.

Examples of One Person, One Power include:

Anime and Manga

Code Geass
Aphorism
The Contractors in Darker than Black follow this rule strictly. Some of them even lack Required Secondary Powers.
Witch Hunter Robin
In One Piece, there appears to be only one of each Devil Fruit. Averted with Haki, though.
Bleach
NEEDLESS
The Espers from A Certain Magical Index and its spin-off A Certain Scientific Railgun always only have one power - but the versatility of this power can vary. Railgun controls electromagnetism which allows for Lightning Can Do Anything, while Meltdowner essentially creates plasma which is mainly only useful for destructive purposes. At the beginning of the story, Combo-Platter Powers - the mythical "Dual Skill" and "Multiskill" - are believed to be impossible in this paradigm.

Comic Books

The "White Event" in The New Universe played this fairly straight.
Lanfeust de Troy
Power Pack. Subverted in that the Power kids' powers weren't eternally affixed to them, and they frequently swapped their powers around. And in at least one story set in the future of the Marvel Universe, Katie Power was the last living member of the Pack and possessed all four of the Pack's power sets.

Film

Push
Sky High. While Nurse Specs mentions that it's possible for the child of two superhumans to inherit both of their parents' powers, the implication is that it's damned rare, and everyone else's reactions to Will's not only being super-strong but being able to fly implies that most people only have one power.

Literature

The Nightfall books by Mickey Zucker Reichert feature individuals with single "natal talents".
The supers in Those Who Walk in Darkness and What Fire Cannot Burn by John Ridley all have strictly one power each. This became a major plot point in the latter book.
The Xanth books fit this trope.
The titular Gracelings each have one Grace (which is either a skill at which they are superhumanly proficient or an outright superpower.)
The psychic children of the Gotterelektrongruppe in Ian Tregillis' Bitter Seeds.
The Magicals/Actives of Hard Magic all have one power each, albeit with Required Secondary Powers included.
Gone (novel) by Michael Grant features several superpowered kids with a single power each.
In Shadow Ops, "Latents" (humans who develop magical powers) only develop within one specific "school" of magic. Some of these are limited, i.e. pyromancers can only create fire. Others are impressively diverse, such as terramancers, who can control earth, shape and grow plants, and mind-control animals.

Live Action TV

The 4400
Heroes has this for all the heroes. It's just that a couple of them have the single power of stealing/copying other people's powers.
Mutant X does this with the exception of Gabriel Ashlocke, who had every power.
Misfits subverts this. Although everyone seems to get one power, at least two people were given two powers in the storm, and Seth can give customers multiple powers with his ability.
Alphas

Web Original

In the Global Guardians PBEM Universe, this is how most superpowers work... while a person might have several abilities, they are usually just variants and special-purpose uses of their one superpower.
Most descendant powers work this way in The Descendants. There are exceptions, but they're rare enough that the fact that magic users don't conform to this rule is an immediate tip off that something is stranger than usual about them.
- The series gets away with Flying Brick, Flying Firepower, and Flight, Strength, Heart by lumping them into either Elemental Powers of a kind of Involuntary Shapeshifting called "protomorphism".