  We give improved estimates for the size of the singular set of minimizing
capillary hypersurfaces: the singular set is always of codimension at least
$4$, and this estimate improves if the capillary angle is close to $0$,
$\frac{\pi}{2}$, or $\pi$. For capillary angles that are close to $0$ or $\pi$,
our analysis is based on a rigorous connection between the capillary problem
and the one-phase Bernoulli problem.
