Such a set $S$ is a closed, discrete subset of $X$. A space $X$ with the property that whenever $S\subseteq X$ is closed and discrete, there is a family $\{V_s:s\in S\}$ of pairwise disjoint open sets such that $s\in V_s$ for each $s\in S$ is said to be collectionwise Hausdorff. This is a weakening of the property of collectionwise normality, which says that every discrete family of closed subsets (not just points) can be expanded to pairwise disjoint open sets. Every paracompact space is collectionwise normal and therefore collectionwise Hausdorff, and every metric space is paracompact, so every metric space is collectionwise Hausdorff. However, there are some quite nice spaces that are not collectionwise Hausdorff. One is Bing’s so-called Example G, which is normal but not collectionwise Hausdorff.
A simpler example that isn’t quite so nice can easily be obtained from the Sorgenfrey line, whose underlying set is $\Bbb{R}$ and whose topology has as a base the set of left-closed, right-open intervals $[a,b)$ such that $a,b\in\Bbb{R}$ with $a. Call this space $S$. $S$ is a GO-space, or generalized ordered space, a property equivalent to (and sometimes defined as) being a subspace of a LOTS (linearly ordered topological space). Every GO-space is hereditarily collectionwise normal, and $S$ is in addition first countable and Lindelöf, so it’s a pretty nice space. Now let $X=S\times S$, sometimes called the Sorgenfrey plane. $X$ is still pretty nice: it’s first countable and completely regular, for instance. However, $X$ is neither normal nor collectionwise Hausdorff: the antidiagonal, $\Delta=\{\langle x,-x\rangle:x\in S\}$, is a closed discrete set that can’t be expanded to a collection of pairwise disjoint open sets, and $Q=\{\langle x,-x\rangle\in\Delta:x\in\Bbb{Q}\}$ and $\Delta\setminus Q$ are disjoint closed sets that can’t be separated by disjoint open sets.