Psychic

A psychic is an individual that possesses supernatural or paranormal powers.

Fallout

 * The Master attempted to create psychic creatures, but most of the experiments were failures resulting in insanity. Four individuals in the Cathedral successfully gained the abilities, referred to as psychics. These characters include Wiggum, Lucy, Moore, and Gideon. A device known as a psychic nullifier was utilized to block incoming thoughts, including those from the area of revulsion. The ability to resist the Master's psychic attacks is represented in the SPECIAL system by the Mental Block perk.
 * Chuck in Adytum will offer to read the Vault Dweller's tarot cards.
 * Tycho mentions that he does not believe in psychics or telepathic abilities.

Fallout 2

 * Hakunin, the Arroyo shaman is able to speak to the Chosen One in their dreams.
 * Melchior is a super mutant who has the ability to command creatures such as deathclaws, fire geckos, and mole rats.

Fallout Tactics
The Beastlords have the ability to command creatures.

Fallout 3

 * A member of the Treeminders named Bloomseer Poplar can share what she sees in the Lone Wanderer's future.
 * Professor Calvert underwent an experimental procedure to extract his brain in an attempt to extend his longevity. Because of experiments involving bio med gel composition, he has the ability to project his thoughts into the minds of others, and wishes to use this ability to achieve "psychic domination."

Fallout: New Vegas

 * The Forecaster is a child who can provide visions of the future. If he utilizes this ability too much, he suffers from headaches and wears a psychic nullifier, which he refers to as his "medicine," to avoid the side effects.

Fallout 4

 * Mama Murphy is a woman who can foresee events after taking chems, an ability she calls "the Sight." She can share these visions with the Sole Survivor.
 * Lorenzo Cabot has powers stemming from an ancient artifact that he discovered during an archaeological dig in the Rub' al Khali. He describes the abilities he has gained, such as exploring the memories of the ancients of Ubar, communing with those who founded our civilization. He also gained a number of other supernatural abilities, such as halted aging, telekinesis, and increased strength and resistances. Some of these abilities can be artificially transferred to others, albeit temporarily, through the consumption of special serum derived from Lorenzo's blood. This has been the case for his family and, may potentially be so for the Sole Survivor.

Fallout 76

 * The Flatwoods monster is able to warp the mind "of all but the most iron-willed."
 * The Interloper sent visions to Jeff Lane, beckoning him to Lucky Hole Mine in order to become its conduit.
 * The raider punk mentions that some peers in his network are psychics.

Behind the scenes
When asked about paranormal and supernatural content in the Fallout series, Joshua Sawyer stated that he feels that "a li'l bit is okay," but prefers if "it feels alien and remains more or less unexplained."