  Let $D$ be an integral domain and $L$ be a field containing $D$. We study the
isolated points of the Zariski space $\mathrm{Zar}(L|D)$, with respect to the
constructible topology. In particular, we completely characterize when $L$ (as
a point) is isolated and, under the hypothesis that $L$ is the quotient field
of $D$, when a valuation domain of dimension $1$ is isolated; as a consequence,
we find all isolated points of $\mathrm{Zar}(D)$ when $D$ is a Noetherian
domain. We also show that if $V$ is a valuation domain and $L$ is
transcendental over $V$ then the set of extensions of $V$ to $L$ has no
isolated points.
