  Under natural hypotheses we give an upper bound on the dimension of families
of singular curves with hyperelliptic normalizations on a surface S with p_g(S)
>0 via the study of the associated families of rational curves in Hilb^2(S).
  We use this result to prove the existence of nodal curves of geometric genus
3 with hyperelliptic normalizations, on a general K3 surface, thus obtaining
specific 2-dimensional families of rational curves in its Hilbert square. We
describe two infinite series of examples of general, primitively polarized K3's
such that their Hilbert squares contain a IP^2 or a threefold birational to a
IP^1-bundle over a K3.
  We discuss some consequences on the Mori cone of the Hilbert square of a
general K3.
