Summary: We consider the form of the eigenvalue counting function \(\rho\) for Laplacians on p.c.f. self-similar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set \(K\) the function \(\rho(x)= O(x^{d_s/2})\) as \(x\to\infty\), 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 \(\rho(x)x^{-d_s/2}\) does not converge as \(x\to\infty\). 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.