The Disk Substructures at High Angular Resolution Project (DSHARP) used the Atacama Large Millimeter/submillimeter Array (ALMA) to map the 1.25mm continuum of protoplanetary disks at a spatial resolution of ~5au. We present a systematic analysis of annular substructures in the 18 single-disk systems targeted in this survey. No dominant architecture emerges from this sample; instead, remarkably diverse morphologies are observed. Annular substructures can occur at virtually any radius where millimeter continuum emission is detected and range in widths from a few astronomical units to tens of astronomical units. Intensity ratios between gaps and adjacent rings range from near-unity to just a few percent. In a minority of cases, annular substructures coexist with other types of substructures, including spiral arms (3/18) and crescent-like azimuthal asymmetries (2/18). No clear trend is observed between the positions of the substructures and stellar host properties. In particular, the absence of an obvious association with stellar host luminosity (and hence the disk thermal structure) suggests that substructures do not occur preferentially near major molecular snowlines. Annular substructures like those observed in DSHARP have long been hypothesized to be due to planet-disk interactions. A few disks exhibit characteristics particularly suggestive of this scenario, including substructures in possible mean-motion resonance and "double gap" features reminiscent of hydrodynamical simulations of multiple gaps opened by a planet in a low-viscosity disk.