  We introduce a diffused interface formulation of the Plateau problem, where
the Allen--Cahn energy $\mathcal{AC}_\varepsilon$ is minimized under a volume
constraint $v$ and a spanning condition on the level sets of the densities. We
discuss two singular limits of these Allen--Cahn Plateau problems: when
$\varepsilon\to 0^+$, we prove convergence to the Gauss' capillarity
formulation of the Plateau problem with positive volume $v$; and when
$\varepsilon\to 0^+$, $v\to 0^+$ and $\varepsilon/v\to 0^+$, we prove
convergence to the classical Plateau problem (in the homotopic spanning
formulation of Harrison and Pugh). As a corollary of our analysis we resolve
the incompatibility between Plateau's laws and the Allen--Cahn equation implied
by a regularity theorem of Tonegawa and Wickramasekera. In particular, we show
that Plateau-type singularities can be approximated by energy minimizing
solutions of the Allen--Cahn equation with a volume Lagrange multiplier and a
transmission condition on a spanning free boundary.
