A crucial issue in capillarity-type problems is understanding the behavior of solutions near singular points in the boundary of the container. In the special case of the relative perimeter functional, we prove that the closure of the internal boundary of a perimeter almost-minimizer in a 3-dimensional convex container Ω cannot contain vertices of Ω. As a byproduct of our study, we also prove a boundary monotonicity formula for perimeter almost-minimizers within a convex container of any dimension, without any extra regularity of the boundary but a suitable closeness to its tangent cones. This is joint work with Giacomo Vianello (Ph.D. UniTN).