  It is known that every closed oriented 3-manifold is homology cobordant to a
hyperbolic 3-manifold. By contrast we show that many homology cobordism classes
contain no Seifert fibered 3-manifold. This is accomplished by determining the
isomorphism type of the rational cohomology ring of all Seifert fibered
3-manifolds with no 2-torsion in their first homology. Then we exhibit families
of examples of 3-manifolds (obtained by surgery on links), with fixed linking
form and cohomology ring, that are not homology cobordant to any Seifert
fibered space (as shown by their rational cohomology ring). These examples are
shown to represent distinct homology cobordism classes using higher Massey
products and Milnor's u-invariants for links.
