This question is based on General Literature Principles and Literary Appreciation. When a story is told in a song and passed down by word of mouth, it is ca...

A story told in a song and passed down by word of mouth is called a ballad. Ballads are narrative poems or songs that tell a story - often about love, adventure, heroism, or tragedy - and have traditionally been transmitted orally from one generation to the next. The two defining features that match the question precisely are the narrative element ("a story is told") and the oral transmission ("passed down by word of mouth").

A sonnet is a 14-line poem with a fixed rhyme scheme (Petrarchan or Shakespearean); it is typically lyric rather than narrative and is a written form, not an oral one. An epic is a long narrative poem about grand heroic deeds (such as Homer's Iliad or Milton's Paradise Lost); while epics have oral roots, they are defined by their length, elevated style, and heroic subject matter rather than by being sung and passed down informally. "A singer's story" is not an established literary term.