For example, it is sufficient that $f\in C^1$ and the set $\lbrace x | f^\prime(x) = 0 \rbrace$ has measure zero.

For example, it is sufficient that $f\in C^1$ and the set $\lbrace x | \nabla f(x) = 0 \rbrace$ has measure zero.

To prove this, note that this is true locally, in a neighborhood of each point where $\nabla f \neq 0$, due to the implicit function theorem. Now the claim follows from the fact that $f^{-1}[A] \subset Z \cup \bigcup_n (U_n \cap f^{-1}[A])$, where $Z = \lbrace \nabla f = 0 \rbrace$ and $(U_n)$ is a countable covering of $\mathbb{R}^m \setminus Z$ by neighborhoods for which the implicit function theorem applies.