  We investigate criteria ensuring that a one-relator group $G$ contains a
right-angled Artin subgroup $A(\Gamma)$, corresponding to a finite graph
$\Gamma$. In particular, we prove that if the positive submonoid $T(\Gamma)$,
of $A(\Gamma)$, embeds into $G$ then so does all of $A(\Gamma)$, unless
$\Gamma$ is totally disconnected. As by-products of our methods we obtain
characterisations of one-relator groups that have property $P_{nai}$ and that
are $C^*$-simple.
