  In this article, we define Perelman's functionals on manifolds with
non-isolated conical singularities by starting from a spectral point of view
for the Perelman's $\lambda$-functional. (Our definition of non-isolated
conical singularities includes isolated conical singularities.) We prove that
the spectrum of Schr\"odinger operator $-4\Delta + R$ on manifolds with
non-isolated conical singularities consists of discrete eigenvalues with finite
multiplicities, provided that scalar curvatures of cross sections of cones have
a certain lower bound. This enables us to define the $\lambda$-functional on
these singular manifolds, and further, to prove that the infimum of
$W$-functional is finite, with the help of some weighted Sobolev inequalities.
Furthermore, we obtain some asymptotic behavior of eigenfunctions and the
minimizer of the $W$-functional near the singularity, and a more refined
optimal partial asymptotic expansion for eigenfunctions near isolated conical
singularities. We also study the spectrum of $-4\Delta + R$ and Perelman's
functionals on manifolds with more general singularities, i.e. the
$r^{\alpha}$-horn singularities which serve as prototypes of algebraic
singularities.
