Powers

Chi
It is possible for a small percentage of humans who've been stung or swallowed by a Dolly Mosquito to awaken supernatural powers.

Once this has occured, the individual will be able to tap into an enhanced version of their "Chi" (or more simply their "Life Force") which they're capable of controlling, either projecting outside of their bodies or internalizing it so as to enhance their natural physical abilities.

In order to properly utilzie this ability, one typically requires both an understanding of the power itself and special kind of training.

Phantasm
Described an enhanced version of chi, "Phantasm" is an energy source connected to the user's own will. Through proper training (e.g. "Qi Gong" and/or "Chi Manipulation") a person is capable of putting the ability into practical use.

Although not magic, it can give the user various superhuman abilities; most notably enhanced strength, speed, endurance, and healing.

There are three breeds of Phantasm users:
 * Anagami - Anagami seem to be the most prevalent kind of Phantasm users. They have direct control over their Phantasm, being capable of channeling it either offensively by focusing it around their limbs or defensively by wrapping their body in it.
 * Arhat - There are only two currently known Arhat-type Phantasm users, both of whom have wildly different abilities:
 * Iruma Ikaruga is the first known to the reader, and his ability is considerably weaker in terms of variety. He is able to gain superhuman strength and enhanced regeneration with the help of blood-like "Insects" which utilize tissue from other parts of his body to repair his injures, giving him a form of quasi-immortality limited only by how far he's willing to let his body degenerate, although simultaneously a dark inner self (as seen in many of Banira's "Soul Dives") consumes a part of his soul each time he uses this power. Due to this power being in such an unnurtured state, having been used only in times of extreme rage or desperation, Iruma has little control when he uses it.
 * Kamuragi Sekka is the second introduced and, later on, is revealed to be the first in history as explained by himself. His abilities go far beyond Iruma, being able to form his Phantasm into beast-like figures which act like real matter. As a side note, since Sekka was capable of implanting the Insects that give regeneration into an Anagami user, it's entirely possible the regeneration power isn't restricted to just Arhats, but can also be gained by any Phantasm users.
 * Sotapanna - Unlike the other two categories, the full extent of what a Sotapanna-type can do is unknown as there has only beem single known user introduced so far, Yuu Banira whose ability, "Soul Dive", seems so unique in its effect that it's highly probable other Sotapannas have different abilites.[speculation]

Insects
Some Phantasm users have the Dolly Insects inside of them giving them additional powers. The Insects feed on Phantasm, allowing them to enhance the bodies physical capabilities even further then Phantasm alone. In addition, the Insects can grant a host an enhanced healing factor so great it can restore severed limbs. Not all Phantasm users seem to have the Insects in them, but seem to be able to attain them by swallowing a live Dolly Mosquito or by transferring them from one Phantasm user (ex. Kamuro). While it is possible for someone with the Insects to be able to control and manipulate Phantasm, it isn't a requirement as is demonstrated with Iruma as he had power of the Insects before he even learned how to control Phantasm.

Those with powers are the 'Awakened ' born from the sacrifice of the rest of the world, according to Madoka. Each and every Awakened seems to have their own unique abilities as well, those being:

Iruma has the ability to pass through Dolly and Dollhouse barriers, a property he can bestow onto anything he's touching as well such as weapons, vehicles, and even other people. He also possess the ability to regenerate by pulling tissue from other parts of his body in order to heal the injured portions. Whenever he goes berserk, Iruma can take on a Dolly-like form which grants him enough strength and speed to beat a Plushie-type Dolly.

Kyujin has th ability to break through barriers by focusing all his power into a single point and then channeling a huge burst of energy into the targetted area after contact, granting him a one-hit kill move. This power is a stronger version of the regular Qi Gong and Chi Manipulation.

Banira has the ability to sense the feelings of other people as well as see visions of their pasts.

Kumano has the ability to create golems out of matter.

The ability to infuse his Phantasm into weapons, such as a sword or a gun, to augment their destructive power; being able to increasing the sharpness of the former whilst using the latter to shoot his Phantasm as bullets.

The ability to steal another's Phantasm ability if his Phantasm is greater.

The ability to take over an individual's mind.

The ability to mimic diseases and manufacture their own poisons as well. However, this Phantasm is also capable of harming the user.

The ability to increase the size of the user, dramatically increasing their strength and durability.

The ability to control animals with Phantasm. However, its not understood if the user actually fully manipulating the animals or if they're merely directing the animals whilst infusing them with power. This ability is also capable of taking control of Dollies.