I will discuss regularity of solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. The work of E. Giusti and others, establishes a very general existence theory of solutions with "unattained Dirichlet data" by minimizing an appropriately defined functional, which includes information about the boundary data. We can naturally associate to such a solution a current, which inherits a natural minimizing property. The main goal is to show that its support is a $C^{1,alpha}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class $C^{1,alpha}$. Furthermore, as a consequence, I will discuss some interesting results about the trace of such a solution; in particular for a large class of boundary data with jump discontinuities, the trace has a jump discontinuity along which it attaches to the vertical part of the prescribed boundary.