  We determine explicitly the irreducible components of the singular locus of
any Schubert variety for GL_n(K), K being an algebraically closed field of
arbitrary characteristic. We also describe the generic singularities along
these components.
  The case of covexillary Schubert varieties was solved in an earlier work of
the author [Ann. Inst. Fourier vol. 51 fasc. 2 (2001), 375-393]. Here, we first
exhibit some irreducible components of the singular locus of X_w, by describing
the generic singularity along each of them. Let S_w be the union of these
components. As mentioned above, the equality S_w = Sing X_w is known for
covexillary varieties, and we base our proof of the general case on this
result. More precisely, we study the exceptional locus of certain
quasi-resolutions of a non-covexillary Schubert variety X_w, and we relate the
intersection of these loci to S_w. Then, by induction on the dimension, we can
establish the equality.
