We introduce simplified models for enhancements in the matter power spectrum at small scales and study their implications for dark matter substructure and gravitational observables. These models capture the salient aspects of a variety of early Universe scenarios that predict enhanced small-scale structure, such as axionlike particle dark matter, light vector dark matter, and epochs of early matter domination. We use a model-independent, semianalytic treatment to map bumps in the matter power spectrum to early forming subsolar mass dark matter halos and estimate their evolution, disruption, and contribution to the substructure of clusters and galaxies at late times. We discuss the sensitivity of gravitational observables, including pulsar timing arrays and caustic microlensing, to both the presence of bumps in the power spectrum and variations in their basic properties.