Disordered systems assembled from a predetermined distribution of constituents represent well-defined materials with an enormous variety of possible microstructures. Material properties will vary with the structure, and recent studies have shown that rare states can have extraordinary properties such as mechanical stability much greater than that of a typical structure. Diagnosing and/or constructing such special states on the complex energy landscape of metastable states (or inherent structures) is a challenge that can be met by following morphological indicators predictive of mechanical energy and desired mechanical properties. For systems in two dimensions, we show that different indicators are advantageous for domain systems (space-filling polygonal constituents like foam bubbles and biological cells in epithelia) on the one hand and for packings (soft disks with a variety of interactions) on the other hand. Strong correlations then allow predictions of mechanical energy, and furthermore of mechanical stability, using just snapshots of the system morphology. For domain systems, we make use of the joint size and topology distributions, while for packings, an angle order parameter gives accurate results and furthermore inspires a new and vastly improved method for constructing mechanically special states. This work has potential impact in applications ranging from tissue diagnostics and morphogenesis to the mechanics of unjamming and the stability of amorphous glassy materials.