We present a programmable droplet-based microfluidic device that combines the reconfigurable flow-routing capabilities of integrated microvalve technology with the sample compartmentalization and dispersion-free transport that is inherent to droplets. The device allows for the execution of user-defined multistep reaction protocols in 95 individually addressable nanoliter-volume storage chambers by consecutively merging programmable sequences of picoliter-volume droplets containing reagents or cells. This functionality is enabled by "flow-controlled wetting," a droplet docking and merging mechanism that exploits the physics of droplet flow through a channel to control the precise location of droplet wetting. The device also allows for automated cross-contamination-free recovery of reaction products from individual chambers into standard microfuge tubes for downstream analysis. The combined features of programmability, addressability, and selective recovery provide a general hardware platform that can be reprogrammed for multiple applications. We demonstrate this versatility by implementing multiple single-cell experiment types with this device: bacterial cell sorting and cultivation, taxonomic gene identification, and high-throughput single-cell whole genome amplification and sequencing using common laboratory strains. Finally, we apply the device to genome analysis of single cells and microbial consortia from diverse environmental samples including a marine enrichment culture, deep-sea sediments, and the human oral cavity. The resulting datasets capture genotypic properties of individual cells and illuminate known and potentially unique partnerships between microbial community members.