  Motivic Chern and Hirzebruch classes are polynomials with K-theory and
homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson
classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas
to compute the motivic Chern and Hirzebruch classes of Grassmannian and
vexillary degeneracy loci. We apply our results to obtain the Hirzebruch
$\chi_y$-genus of classical and one-pointed Brill-Noether varieties, and
therefore their topological Euler characteristic, holomorphic Euler
characteristic, and signature.
