  We prove a quantitative estimate on the number of certain singularities in
almost minimizing clusters. In particular, we consider the singular points
belonging to the lowest stratum of the Federer-Almgren stratification (namely,
where each tangent cone does not split a $\R$) with maximal density. As a
consequence we obtain an estimate on the number of triple junctions in
$2$-dimensional clusters and on the number of tetrahedral points in $3$
dimensions, that in turn implies that the boundaries of volume-constrained
minimizing clusters form at most a finite number of equivalence classes modulo
homeomorphism of the boundary, provided that the prescribed volumes vary in a
compact set. The method is quite general and applies also to other problems:
for instance, to count the number of singularities in a codimension 1
area-minimizing surface in $\R^8$.
