  Drinfeld orbifold algebras are a type of deformation of skew group algebras
generalizing graded Hecke algebras of interest in representation theory,
algebraic combinatorics, and noncommutative geometry. In this article, we
classify all Drinfeld orbifold algebras for symmetric groups acting by the
natural permutation representation. This provides, for nonabelian groups,
infinite families of examples of Drinfeld orbifold algebras that are not graded
Hecke algebras. We include explicit descriptions of the maps recording
commutator relations and show there is a one-parameter family of such maps
supported only on the identity and a three-parameter family of maps supported
only on 3-cycles and 5-cycles. Each commutator map must satisfy properties
arising from a Poincar\'{e}-Birkhoff-Witt condition on the algebra, and our
analysis of the properties illustrates reduction techniques using orbits of
group element factorizations and intersections of fixed point spaces.
