We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined using orderings of the cells of a pseudomanifold. In addition to defining and finding combinatorial minimal surfaces, from thin orderings, we derive invariants of even-dimensional closed simplicial pseudomanifolds called width and trunk. We study additivity properties of these invariants under connected sum and prove theorems analogous to those in knot theory and 3-manifold theory.