The Stacks project

Definition 28.14.1. Let $X$ be a locally Noetherian scheme. The regular locus $\text{Reg}(X)$ of $X$ is the set of $x \in X$ such that $\mathcal{O}_{X, x}$ is a regular local ring. The singular locus $\text{Sing}(X)$ is the complement $X \setminus \text{Reg}(X)$, i.e., the set of points $x \in X$ such that $\mathcal{O}_{X, x}$ is not a regular local ring.