Rigidity is the ability of a system to resist imposed stresses before ultimately undergoing failure. However, disordered materials often contain both rigid and floppy subregions that complicate the utility of taking system-wide averages. I will talk about 3 frameworks capable of connecting the internal structure of disordered materials to their rigidity and/or failure under loading, and describe how my collaborators and I have applied these frameworks to laboratory data on laser-cut lattices and idealized granular materials. These are, in order of increasing physics content: (1) centrality within an adjacency matrix describing its connectivity, (2) Maxwell constraint counting on the full network of frictional contact forces, and (3) the vibrational modes of a synthetic dynamical matrix (Hessian). The first two rely primarily on topology, and the second two contrast the utility of considering interparticle forces (Coulomb failure) vs. the energy landscape. All three methods, while successfully elucidating the origins of rigidity and brittle vs. ductile failure, also provide interesting counterpoints regarding how much information is enough to make predictions.