The Master appeared to at least possess some psychic ability when he could sense a powerful psychic force caused by the astral projection of Billy Palmer (though it was possible he was sensing the Hellmouth having resided so close to it underground, it being the cause of the problem).
Drusilla passively received vivid visions that contained possible glimpses of the future. With them, she sensed Angelus' presence when Darla introduced him to her and she knew what he would do to her and those around her, allowing her to prompt her family to leave. She also may have seen some of the upcoming events of Buffy's junior year in high school, when saying to Angel during their reunion "Poor little thing. She has no idea what's in store" and knew what was happening then and there was "just the beginning".
Doyle received prophetic visions from The Powers That Be, which included images of people in peril, names, places in which evil was present or threats the Powers wanted to be dealt with. These visions also caused Doyle a great pain, though his half-demon physiology prevented his brain from suffering real damage.
After Doyle's death, the visions were passed on to Cordelia. Like Doyle's had, the visions usually consisted of ambiguous imagery of forthcoming attacks on innocents or various demonic disasters. Cordelia used this imagery to help Angel prevent them from happening.
The Holy Triumvirate were a trio of blind children from different parts of the world who, according to the Scrolls of Aberjian, would one day be able to see into the heart of all things and pose a threat to Wolfram & Hart.
Lorne was capable of reading people's auras, as well as their futures, while they were singing, humming or whistling. He used this power to set them on their path in life. Lorne could also read their thoughts and emotions when he did this. Although he had an ability to read basic details about auras even without the need for singing and could read detailed information about people who were in extreme emotional distress without them singing. His powers did not require conscious effort, as he was capable of sensing Holtz's plans to blow up Caritas just by hearing him whistle, as well as Wesley's plans to kidnap Connor by hearing him hum a lullaby. Regardless, his ability was not infallible, and could be fooled by magic and substances such as calendula.
Much like Cordelia, Tammy was a human who had the visions in 1630. She had the visions for a year and was branded a witch by her town's fathers, but her last vision caused enough brain damage until they blew out the back of her head, killing her, before they could burn her at the stake.
Cassie's precognitive abilities manifested themselves not in the form visions of premonitions, but in the form of vague knowledge and intuition. Her abilities granted her vague knowledge of future events, but not their causes, thus leaving her nearly helpless to prevent them. Some of her visions included knowledge of her death, her friend's grade and Buffy Summers spilling coffee onto her sweater.