  In the 1970s O. Zariski introduced a general theory of equisingularity for
algebroid and algebraic hypersurfaces over an algebraically closed field of
characteristic zero. His theory builds up on understanding the dimensionality
type of hypersurface singularities, notion defined recursively by considering
the discriminants loci of successive "generic" corank 1 projections. The theory
of singularities of dimensionality type 1, that is the ones appearing
generically in codimension 1, was developed by Zariski in his foundational
papers on equisingular families of plane curve singularities. In this paper we
completely settle the case of dimensionality type 2, by studying Zariski
equisingular families of surfaces singularities, not necessarily isolated, in
the three-dimensional space.
