  We study the reach (in the sense of Federer) of the natural isometric
embedding $X\hookrightarrow W_p(X)$ of $X$ inside its $p$-Wasserstein space,
where $(X,\operatorname{dist})$ is a geodesic metric space. We prove that if a
point $x\in X$ can be joined to another point $y\in X$ by two minimizing
geodesics, then $\operatorname{reach}(x, X\subset W_p(X)) = 0$. This includes
the cases where $X$ is a compact manifold or a non-simply connected one. On the
other hand, we show that $\operatorname{reach}(X\subset W_p(X)) = \infty$ when
$X$ is a CAT(0) space. The infinite reach enables us to examine the regularity
of the projection map. Furthermore, we replicate these findings by considering
the isometric embedding $X\hookrightarrow W_\vartheta(X)$ into an
Orlicz--Wasserstein space, a generalization by Sturm of the classical
Wasserstein space. Lastly, we establish the nullity of the reach for the
isometric embedding of $X$ into $\operatorname{Dgm}_\infty$, the space of
persistence diagrams equipped with the bottleneck distance.
