A fable was a type of story. One notable Earth fable was The Little Mermaid. (DS9: "Melora")
During his childhood, Christopher Pike heard a fable in which a person was punished in a hellish environment. Decades later, as a prisoner of the Talosians, he was forced to experience this event as punishment for his disobedience. The Talosian warned Pike, that from the deeper parts of his mind, there were even more unpleasant things and, if he continued his disobedience, he would experience one of them. (TOS: "The Cage", "The Menagerie, Part II")
In 2268, James T. Kirk dismissed the Klingon rumors Mara had heard about the Federation killing or torturing its prisoners as propaganda and fables. (TOS: "Day of the Dove")
In 2366, Deanna Troi called the Mintakan stories of the Overseer fables (TNG: "Who Watches The Watchers")
According to Tuvok, fables that told of monsters were a byproduct of cultures that emphasized emotions too much. (VOY: "Heroes and Demons") Neelix believed that many had some basis in reality. (VOY: "Dragon's Teeth")