  We consider flat families of reduced curves on a smooth surface S such that
each member C has the same number of singularities of fixed singularity types
and the corresponding (locally closed) subscheme H of the Hilbert scheme of S.
We are mainly concerned with analytic resp. topological singularity types and
give a sufficient condition for the smoothness of H (at C). Our results for
S=P^2 seem to be quite sharp for families of cuves of small degree d.
