  In 1978 Durfee conjectured various inequalities between the signature and the
geometric genus of a normal surface singularity. Since then a few counter
examples have been found and positive results established in some special
cases.
  We prove a `strong' Durfee--type inequality for any smoothing of a Gorenstein
singularity, provided that the intersection form the resolution is unimodular,
and the conjectured `weak' inequality for all hypersurface singularities and
for sufficiently large multiplicity strict complete intersections. The proofs
establish general inequalities valid for any normal surface singularity.
