  We present a combinatorial and computational commutative algebra methodology
for studying singularities of Schubert varieties of flag manifolds.
  We define the combinatorial notion of *interval pattern avoidance*. For
"reasonable" invariants P of singularities, we geometrically prove that this
governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties
are globally not P. The prototypical case is P="singular"; classical pattern
avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is
insufficient in general.
  Our approach is analyzed for some common invariants, including
Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness,
extending [Woo-Yong'05]; the description of the singular locus (which was
independently proved by [Billey-Warrington '03], [Cortez '03],
[Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted.
  Our methods are amenable to computer experimentation, based on computing with
*Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using
Macaulay 2. This feature is supplemented by a collection of open problems and
conjectures.
