Singularities of the Isospectral Hilbert Scheme

We study the singularities of the isospectral Hilbert scheme $B^n$ of $n$ points over a smooth algebraic surface and we prove that they are canonical if $n\le 5$, log-canonical if $n\le 7$ and not log-canonical if $n\ge 9$. We describe as well two explicit log-resolutions of $B^3$, one crepant and the other ${\mathfrak{S}}_3$-equivariant.