Who's is a shortened form of "who is": "Who's ahead in the office pool?" Whose is the possessive case of the pronoun who: "Whose shoes are these?" Some grammarians formerly insisted that whose should be applied only to persons: "The car the body of which needed paint" rather than "The car whose body needed paint." Both common sense and the fact that which has no possessive form of its own have succeeded in discarding this "rule." Therefore, continue to sing about the flag "Whose broad stripes and bright stars, through the perilous fight, / O'er the ramparts we watched were so gallantly streaming."
Dictionary of problem words and expressions. Harry Shaw. 1975.