Enhanced inverter-based controls are considered for microgrids, which use additional actuation beyond a droop-like term. Shaping of the microgrid’s small-signal dynamics using such enhanced controls is posed as a structural controllability problem. A graph-theoretic characterization of structural controllability is obtained, in terms of the concept of zero-forcing sets. Two benchmark test systems are used to illustrate the selection of locations where enhanced controls should be applied, based on the graph-theoretic analysis. These examples indicate that small-signal characteristics can be shaped using a relatively small number of enhanced controls.