The lyric “My fair...” is completed by “lady” in the familiar version of “London Bridge Is Falling Down.”
The phrase appears at the end of the opening stanza: the bridge is falling down, and the refrain addresses a “fair lady.” “Fair” traditionally means beautiful or pleasing, so the expression is an old-fashioned form of address rather than a reference to a specific modern title.
The rhyme is a traditional English nursery song and singing game. Its earliest English references date to the seventeenth century, while a version close to the modern text was printed in the eighteenth century. The song imagines repeated attempts to repair the bridge with materials such as wood, clay, brick, iron, or steel.
There are many regional and historical variations, and the identity of the fair lady has never been settled. Some theories connect her with a historical woman, a religious figure, or a symbolic character, but none is universally accepted. “Lady” remains the standard answer for the common opening verse.