  The goal of this note is to present some recent results of our research
concerning multiplier ideal sheaves on complex spaces and singularities of
plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on
complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure,
as a special case of which, it turns out to be the so-called Mather-Jacobian
multiplier ideals in the algebro-geometric setting. As applications, we obtain
a reasonable generalization of (algebraic) adjoint ideal sheaves to the
analytic setting and establish some extension theorems on K\"ahler manifolds
from \emph{singular} hypersurfaces. Relying on our multiplier and adjoint
ideals, we also give characterizations for several important classes of
singularities of pairs associated to plurisubharmonic functions.
  Moreover, we also investigate the local structure of singularities of log
canonical locus of plurisubharmonic functions. Especially, in the
three-dimensional case, we show that for any plurisubharmonic function with log
canonical singularities, its associated multiplier ideal subscheme is weakly
normal, by which we give a complete classification of multiplier ideal
subschemes with log canonical singularities.
