I'm looking for the proof of a statement from: https://en.wikipedia.org/wiki/Regular_scheme
Every smooth scheme $X$ is regular, and every regular scheme over a perfect field is smooth.
Here regular means that for every $x \in X$ the stalk $\mathcal{O}_{X,x}$ is regular. The definition of smoothness as introduced here: https://en.wikipedia.org/wiki/Smooth_scheme
is also local therefore we can consider wlog the case $X = Spec(R)$ where $R$ is a $k$-algebra.
Certainly this won't hold in full generality as smooth schemes over a field are necessarily locally of finite type. So the claim would become as follows. Every smooth $k$-scheme is regular, and if $k$ is perfect, then every regular $k$-scheme, locally of finite type, is smooth over $k$. You can find this result in Vakil's Foundations of Algebraic Geometry, Theorem 12.2.10, except that 'locally of finite type' is replaced by 'of finite type', though I expect shouldn't be an issue as smoothness is a local property.