There has been renewed interest in the response properties of disordered materials in recent years. Understanding the "jamming" of hard or soft particles, considered as a zero-temperature phase transition, has led to the development of new techniques in statistical physics. Related phase transitions that occur in disordered elastic networks may hold the key to understanding the highly tunable systems frequently seen in biology. To address these problems, we investigate the scaling properties and extract the universal predictions of a dynamical mean-field theory, the coherent potential approximation (CPA), that describes the phase behavior in inhomogeneous elastic systems quite well (Chapter 2). We make comparisons to measurements of charge density fluctuations in strange metals, which also show featureless response (Chapter 3). We follow up on these universal predictions from the CPA by showing the existence and origin of logarithmic corrections to scaling in two dimensions (Chapter 4). We also perform simulations of an anisotropically diluted version of the triangular lattice and analyze the non-mean-field behavior as a crossover between two distinct universality classes of rigidity transitions as isotropy is broken (Chapter 5). We extract universal scaling functions for Ising models using modern non-perturbative techniques (Chapter 6). Finally, we comment on the topological defects that are possible in the many nematic phases of bent-core liquid crystals (Chapter 7).