  We enumerate smooth and rationally smooth Schubert varieties in the classical
finite types A, B, C, and D, extending Haiman's enumeration for type A. To do
this enumeration, we introduce a notion of staircase diagrams on a graph. These
combinatorial structures are collections of steps of irregular size, forming
interconnected staircases over the given graph. Over a Dynkin-Coxeter graph,
the set of "nearly-maximally labelled" staircase diagrams is in bijection with
the set of Schubert varieties with a complete Billey-Postnikov (BP)
decomposition. We can then use an earlier result of the authors showing that
all finite-type rationally smooth Schubert varieties have a complete BP
decomposition to finish the enumeration.
