Let $f:X\to S$ be a morphism locally of finite type of locally Noetherian schemes. Let $x\in X$. I wonder if the following is a sufficient condition for $f$ being smooth at $x$.
there exists $n\geq0$ such that $\widehat{\mathcal{O}}_{X,x}\cong\widehat{\mathcal{O}}_{S,f(x)}[[t_1,\cdots,t_n]]$.
This seems quite natural, and I would expect an easy-to-cite reference such as the Stacks Project lemma to affirm it.
I refer to Definition 29.34.1 for smoothness of $f:X\to S$ at $x\in X$. It means there are affine open subschemes $x\in\mathrm{Spec}(A)\subseteq X$ and $\mathrm{Spec}(R)\subseteq S$ such that $f(\mathrm{Spec}(A))\subseteq \mathrm{Spec}(R)$ and $R\to A$ is a smooth ring map.
Using the fact that a morphism of finite presentation is smooth iff it is formally smooth, it suffices to check formal smoothness at the point $ x $ and this is easy: If we're given a commutative square as below $\require{AMScd}$ \begin{CD} \operatorname{Spec} A @>{}>> X\\ @VVV @VVV\\ \operatorname{Spec} A' @>{}>> S \end{CD}
with $ A' \rightarrow A $ a square-zero surjection of Artin rings and the horizontal morphisms mapping to points $ x $ and $ s = f(x) $, what is desired is a lift $ \operatorname{Spec} A' \rightarrow X $ making the diagram commute. Because $ A,A' $ are Artin rings, the diagram factors as $\require{AMScd}$ \begin{CD} \operatorname{Spec} A @>{}>> \operatorname{Spec} \widehat{\mathcal{O}_{X,x}} \\ @VVV @VVV\\ \operatorname{Spec} A' @>{}>> \operatorname{Spec} \widehat{\mathcal{O}_{S,s}} \end{CD} so it's enough to get a lift in this commutative square. Equivalently, we just need to lift the morphism $ \widehat{\mathcal{O}_{X,x}} \rightarrow A $ of $ \widehat{\mathcal{O}_{S,s}} $-algebras to a morphism $ \widehat{\mathcal{O}_{X,x}} \rightarrow A' $ but this is obvious from assumption: since $ \widehat{\mathcal{O}_{X,x}} \cong \widehat{\mathcal{O}_{S, s}} [[ z_1, \ldots, z_n ]] $, just lift arbitrarily the images of the $ (z_i) $ in $ A $ to $ A' $.