For finitely generated domains over a base field $k$ of characteristic 0, we have that if $A$ is regular, then both $Der_k \, A$ and the module of Khäler differentials are finitely generated projective (McConnell, Robinson, Noncommutative Noetherian Rings, revised edition, 15.2.11).

Zariski-Lipman's Conjecture says that if $Der_k \, A$ is finitely generated projective (or, in a more modest version, free), then $A$ is regular.

So for this class of algebras (roughly, regular functions on smooth affine varieties), it is expected that $A$ is regular if and only if $Der_k \, A$ is a finitey generated projective module.

Your example does not fit here as smooth functions on a real manifold are not even domains, but I think that the literature on this subject (i.e. Zariski-Lipman's Conjecture) might be a good direction to look.

For finitely generated domains over a base field $k$ of characteristic 0, we have that if $A$ is regular, then both $Der_k \, A$ and the module of Kähler differentials are finitely generated projective (McConnell, Robson, Noncommutative Noetherian Rings, revised edition, 15.2.11).

Zariski-Lipman's Conjecture says that if $Der_k \, A$ is finitely generated projective (or, in a more modest version, free), then $A$ is regular.

So for this class of algebras (roughly, regular functions on smooth affine varieties), it is expected that $A$ is regular if and only if $Der_k \, A$ is a finitey generated projective module.

Your example does not fit here as smooth functions on a real manifold are not even domains, but I think that the literature on this subject (i.e. Zariski-Lipman's Conjecture) might be a good direction to look.