  In a previous paper, the authors introduced a filtration on the ring ${\cal
O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety
defined by arcs on the singularity and called the arc filtration. The
Poincar\'e series of this filtration were computed for simple surface
singularities in the 3-space. Here they are computed for surface singularities
from Arnold's lists including uni- and bimodular ones. The classification of
the unimodular singularities by these Poincar\'e series turns out to be in
accordance with their hierarchy defined by E. Brieskorn using the adjacency
relations. Besides that we give a general formula for the Poincar\'e series of
the arc filtration for isolated surface singularities which are stabilizations
of plane curve ones.
