This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min{g((F-1(Omega(1)), ..., F-h(Omega(h))) + m vertical bar U-i=1(h) Omega(i)vertical bar : Omega(i) subset of D, Omega(i)boolean AND Omega(j) = phi}, where D subset of R-d is a given bounded open set, vertical bar Omega(i)vertical bar is the Lebesgue measure of Omega(i), and m is a positive constant. For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., F-i = lambda(ki).