We prove that the topological type of a normal surface singularity
(
X
,
0
)
(X,0)
provides finite bounds for the multiplicity and polar multiplicity of
(
X
,
0
)
(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
)
(X,0)
. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of
(
X
,
0
)
(X,0)
, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of
(
X
,
0
)
(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.