Let $M^m,N^n$ be smooth manifolds and $f \in C^\infty(M, N)$. Then, for any regular value $y \in N$ we have that $f^{-1}(y)$ is a smooth submanifold of $M$ of dimension $m-n$. Moreover, by Sard's theorem, the set of regular values of $f$ is dense in $N$. However, we do not know anything, in general, about $f^{-1}(y)$ when $y$ is a critical value. Now, suppose that $M, N$ and $f$ are real-analytic. I heard that in this case it can be said something about the structure of the sets $f^{-1}(y)$ also when $y$ is a critical value. In particular, they should be not too bad; that is, something like unions of $C^{1,\alpha}$-curves. The problem is that I cannot find striking references on this fact (althought there is some literature about subanalytic sets/maps).
Q1. What is the "state-of-art" knowledge of the structure of counterimages of critical values of real-analytic mappings?
Q2. Can you provide references?