In this paper we consider the form of the eigenvalue counting function rho for Laplacians on p.c.f. selfsimilar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function rho(x) = O(x(ds/2)) as x --> infinity, for some d(s) > 0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions ('the lattice case') then rho(x)x(-ds/2) does not converge as x --> infinity. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.