Let $A$ be a closed subset of $\mathbb R^n$ :
1) Is it true that for some smooth function $f: \mathbb R^n \to \mathbb R$ , $A=f^{-1}(\{0\})$
2)Is it true that for some smooth function $f: \mathbb R^n \to \mathbb R$ , $A$ is the set of all critical points of $f$ ?
The answer to the first question is "yes", and I guess this is originally due to Whitney. For a proof see Infinitely differentiable function with given zero set?
The answer to the second question is "no" in general for $n \ge 2$. E.g., if $n=2$ and $A$ is the unit circle in the plane, then you can find a regular value $y=f(x)$ where $x$ is contained in the open unit disk. This shows that the component $C$ of $f^{-1}(\{y\})$ containing $x$ is a smooth closed curve contained in the open unit disk. Since $f$ is constant on $C$, it has to have a maximum or minimum in the region enclosed by $C$, which must be a critical point which is not in $A$.
