  Building on work of Saxl, we classify the multiplicity-free permutation
characters of all symmetric groups of degree 66 or more. A corollary is a
complete list of the irreducible characters of symmetric groups (again of
degree 66 or more) which may appear in a multiplicity-free permutation
representation. The multiplicity-free characters in a related family of
monomial characters are also classified. We end by investigating a consequence
of these results for Specht filtrations of permutation modules defined over
fields of prime characteristic. Remark: parallel work of Godsil and Meagher
(arXiv:math/0612567) gives an independent classification of the
multiplicity-free permutation characters of symmetric groups of all degrees.
