  In this paper we extend Y.Eliashberg's $h$-principle to arbitrary generic
smooth maps of smooth manifolds. Namely, we prove a necessary and sufficient
condition for a continuous map of smooth manifolds of the same dimension to be
homotopic to a generic map with a prescribed Thom-Boardman singularity
$\Sigma^I$ at each point. In dimension 3 we rephrase these conditions in terms
of the Stiefel-Whitney classes and the cohomology classes of the given loci of
folds, cusps and swallowtail points.
