This is a collection of remarks about your question. I will treat the question as a question about repersentations in GL(n,R) or GL(n,C).
Then first (simple) remak is that for every finite group the repesenation vairety is smooth. I would not be able to give some other non-trivial (infinite) examples apart from free group and patological examples, say when your group is an infinite simple group, so it does not have representations at all (such group will be non-linear), or combinations of whose.
Second, there is an extencive theory of repersentation variteties of fundamental groups of Kahler manifolds. One refference is Goldman and Milson The deformation theory of representations of fundamental groups of compact Kahler manifolds.
If you track this article on mathscinet, you will get a lot relevant literature. They show in particular that singularites of such varieties are quadratic (theorem 1). Though I did not read this paper, and don't know how readable it is.
It you conisder a simplest infinite group, say fundamental group of a genus $g>1$ surface, its repersentation variety will be singular at the trivial representation (it seems to me this will be quite a common situation) but it will be non-sigular at all points that correspond to irreducible repersentations.
Finally, there are some examples of "terrbile" non-quadratic sinuglarites, for example for repesentation varieties of fundametnal groups of three-dimensional hyperbolic manfiolds (again at the trival repesentation). This is contained in an article of Ghys (cool aricle but in French:) Notice that that 3-dimesnional hyperbolic group is a fundamental group of a Complex but Non-Kahler manifold (this is the main point of the article of Ghys).
Déformations des structures complexes sur les espaces homogènes de SL(2,C). (Reine Angew. Math. 468 (1995), 113-138