Topologically minimal surfaces generalize several well-studied classes of surfaces in 3-manifolds, and provide a topological analogue to geometrically minimal surfaces. We will discuss recent progress in obtaining a normal form for any such surface with respect to a fixed triangulation. This provides striking analogues with results of Colding and Minicozzi, and establishes finiteness results which are crucial to understanding how Heegaard splittings are effected by Dehn surgery.