The flaming sword is no single blade but a wonder worked in the name of R'hllor, the Lord of Light. Thoros of Myr, a red priest with little true faith but great showmanship, learned to soak a sword in wildfire so that it took flame without melting the steel, and dazzled King Robert's tourneys with the trick, frightening green knights and unhorsing them before a blow was struck.
His friend Lord Beric Dondarrion took up a burning blade in earnest, leading the brotherhood without banners through the riverlands with a sword wreathed in fire — and after Thoros called him back from death again and again, the flames came to seem something more than a mummer's trick. Yet the fire is a glamour of faith, not of the metal: an ordinary sword is left scorched and brittle once the flames die.
Status: extant, wielded in the service of R'hllor.