One possible framework in which to study the Plateau problem is by using currents with mod(p) coefficients, for a fixed integer p. This setting allows for minimizing surfaces to exhibit codimension 1 singularities like triple junctions (p=3), and the known regularity theory for general stable minimal surfaces is so far consistent with that for mod(p) minimizers, unlike for area-minimizing integral currents, which exhibit better regularity properties. For mod(p) minimizing hypersurfaces, a reasonably complete characterization of the structure of the interior singular set has recently been established. In this talk, I will discuss joint work in progress with Camillo De Lellis and Paul Minter towards establishing a structural result on the interior singular set when the surface has higher codimension, which is an analogue of that for hypersurfaces. I will emphasize the difficulties that arise here in contrast to the codimension 1 setting.