  We prove that the topological type of a normal surface singularity $(X,0)$
provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$,
as well as for the combinatorics of the families of generic hyperplane sections
and of polar curves of the generic plane projections of $(X,0)$. A key
ingredient in our proof is a topological bound of the growth of the Mather
discrepancies of $(X,0)$, which allows us to bound the number of point blowups
necessary to achieve factorization of any resolution of $(X,0)$ through its
Nash transform. This fits in the program of polar explorations, the quest to
determine the generic polar variety of a singular surface germ, to which the
final part of the paper is devoted.
