  In this paper, we study the Brill-Noether theory of the normalizations of
singular, irreducible curves on a $K3$ surface. We introduce a {\em singular}
Brill-Noether number $\rho_{sing}$ and show that if the Picard group of the K3
surface is ${\mathbb Z} [L]$, there are no $g^r_d$'s on the normalizations of
irreducible curves in $|L|$, provided that $\rho_{sing} <0$.
  We give examples showing the sharpness of this result. We then focus on the
case of {\em hyperelliptic normalizations}, and classify linear systems $|L|$
containing irreducible nodal curves with hyperelliptic normalizations, for
$\rho_{sing}<0$, without any assumption on its Picard group.
