  We study the transverse Lusternik-Schnirelmann category of a Riemannian
foliation on a compact manifold. We obtain a necessary and sufficient condition
when the transverse LS category is finite. We also introduce a variation on the
concept of transverse LS category, the essential transverse category, and show
that this is finite for every Riemannian foliation and coincides with the
transverse category if the latter is finite. Moreover we prove that the
essential transverse category is a lower bound for the number of critical leaf
closures of a basic differentiable function on M.
