  As is well known, the "usual discrepancy" is defined for a normal
Q-Gorenstein variety. By using this discrepancy we can define a canonical
singularity and a log canonical singularity. In the same way, by using a new
notion, Mather-Jacobian discrepancy introduced in recent papers we can define a
"canonical singularity" and a "log canonical singularity" for not necessarily
normal or Q-Gorenstein varieties. In this paper, we show basic properties of
these singularities, behavior of these singularities under deformations and
determine all these singularities of dimension up to 2.
