  This is an extended, renovated and updated report on a joint work which the
second named author presented at the Conference on Algebraic Geometry held at
Saitama University, 15-17 of March, 1995. The main result is an inequality for
the numerical type of singularities of a plane curve, which involves the degree
of the curve, the multiplicities and the Milnor numbers of its singular points.
It is a corollary of the logarithmic Bogomolov-Miyaoka-Yau's type inequality
due to Miyaoka. It was first proven by F. Sakai at 1990 and rediscovered by the
authors independently in the particular case of an irreducible cuspidal curve
at 1992. Our proof is based on the localization, the local Zariski--Fujita
decomposition and uses a graph discriminant calculus. The key point is a local
analog of the BMY-inequality for a plane curve germ. As a corollary, a
boundedness criterium for a family of plane curves has been obtained. Another
application of our methods is the following fact: a rigid rational cuspidal
plane curve cannot have more than 9 cusps.
