Background:
Fix an algebraically closed field $k$ and a group $\mathbb{G}$ which is algebraic over $k.$
Let $H$ be a discrete group.
There is a functor from $k$-algebras to sets which sends $A \to Hom(H, \mathbb{G}(A)).$ This functor is representable. Let $\mathcal{A}_H$ be the representative.
Then $V_H:= Spec(\mathcal{A}_H)$ is called the representation scheme of the group $G.$ In general, this scheme can be non-reduced and singular.
The GIT quotient $X_H:= V_H // \mathbb{G}^{ad}$ is called the character scheme. It is also in general non-reduced and singular.
Finally, there is an open subscheme $X_H^0 \subset X_H$ whose $k$-points consist of irreducible representations.
My question:
I believe that $X_H^0$ can also be singular an non-reduced. However, I don't know any examples. I would be interested in examples with any group $H$, and any algebraic group $\mathbb{G}.$ I would particularly interested in cases where $H$ arises as the fundamental groups of a $3$-manifold.