In 2 dimensions canonical singularities are the du Val singularities, which are quotients by finite subgroups of SL2, so my guess is that quotients by finite subgroups of GL2 that are not in SL2 would be are a good place to look for counterexamples. (However the quotients by some of these subgroups can be nonsingular, so you have to be a bit careful.) Unless I've misunderstood what is going on, most of the cyclic quotient singularities are non-canonical (these are quotients of C2 by a cyclic group acting as (x,y) → (xζ,yζn) for ζ a root of 1).
Reid's paper http://www.warwick.ac.uk/~masda/surf/more/cyclic.pdf gives more details
