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Under natural hypotheses we give an upper bound on the dimension of familiesof 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 genus3 with hyperelliptic normalizations, on a general K3 surface, thus obtainingspecific 2-dimensional families of rational curves in its Hilbert square. Wedescribe two infinite series of examples of general, primitively polarized K3'ssuch that their Hilbert squares contain a IP^2 or a threefold birational to aIP^1-bundle over a K3.  We discuss some consequences on the Mori cone of the Hilbert square of ageneral K3.

On families of rational curves in the Hilbert square of a surface (with an Appendix by Edoardo Sernesi)