In [1-7] the authors proved that there are at least 1879 (and at most 1880) di erent geometric con gurations of singularities of quadratic di erential systems in the plane. This classi cation is completely algebraic and done in terms of invariant polynomials and it is ner than the classi cation of quadratic systems according to the topological classi cation of singularities. The long term project is the classi cation of phase portraits of all quadratic systems under topological equivalence. A rst step in this direction is to obtain the classi cation of quadratic systems under topological equivalence of local phase portraits around singularities. In this paper we extract the local topological information around all singularities from the 1879 geometric equivalence classes. We prove that there are exactly 208 topologically distinct global topological con gurations of singularities for the whole quadratic class. From here the next goal would be to obtain a bound for the number of possible di erent phase portraits, modulo limit cycles.