  In contrast to the familiar (2,2) case, the singularities which arise in the
(0,2) setting can be associated with degeneration of the base Calabi-Yau
manifold {\it and/or}\/ with degenerations of the gauge bundle. We study a
variety of such singularities and give a procedure for resolving those which
can be cured perturbatively. Among the novel features which emerge are models
in which smoothing singularities in the base yields a gauge {\it sheaf}\/ as
opposed to a gauge bundle as the structure to which left moving fermions
couple. Supersymmetric $\sigma$-models with target data being an appropriate
sheaf on a Calabi-Yau space therefore appear to be the natural arena for N=1
string models in four dimensions. We also indicate a variety of singularities
which would require a nonperturbative treatment for their resolution and
briefly discuss applications to heterotic models on K3.
