Topological description of the regular values of a differentiable function

Question

Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms?

In particular, is the set of regular values necessarily non-empty? Are there accumulation points or is it a discrete set? How bad can it get? Do conditions on the support of $f$ (for example: compact support) or on $n$ and $m$ have any effect on the above characteristics for the set of regular values?

I'm very sorry I can't make my question much more precise. References or some orientation as to how to adress these issues would be very much appreciated. The original question motivating this discussion can be found here: Capacity approximations by sets with regular boundary.

Answer 1

The Morse (1939) - Sard (1942) theorem: The set of all singular values of a $C^k$-mapping $f$ is of Lebesgue measure 0 in $\mathbb R^m$ if $k> \max\lbrace 0, n-m\rbrace$. Here a $\mathbb R^m\setminus f(\mathbb R^n)$ consists of regular values.