  In this paper, we study the algebraic symplectic geometry of the singular
moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann
surface $X$ of genus $g$. In particular, we prove that such moduli spaces are
symplectic singularities, in the sense of Beauville [Bea00], and admit a
projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These
results are an application of a recent paper by Bellamy and Schedler [BS16] via
the so-called Isosingularity Theorem.
