Several modern genomic technologies, such as DNA-Methylation arrays, measure spatially registered probes that number in the hundreds of thousands across multiplechromosomes. The measured probes are by themselves less interesting scientifically; instead scientists seek to discover biologically interpretable genomic regions comprised of contiguous groups of probes which may act as biomarkers of disease or serve as a dimension-reducing pre-processing step for downstream analyses. In this paper, we introduce an unsupervised feature learning technique which maps technological units (probes) to biological units (genomic regions) that are common across all subjects. We use ideas from fusion penalties and convex clustering to introduce a method for Spatial Convex Clustering, or SpaCC. Our method is specifically tailored to detecting multi-subject regions of methylation, but we also test our approach on the well-studied problem of detecting segments of copy number variation. We formulate our method as a convex optimization problem, develop a massively parallelizable algorithm to find its solution, and introduce automated approaches for handling missing values and determining tuning parameters. Through simulation studies based on real methylation and copy number variation data, we show that SpaCC exhibits significant performance gains relative to existing methods. Finally, we illustrate SpaCC's advantages as a pre-processing technique that reduces large-scale genomics data into a smaller number of genomic regions through several cancer epigenetics case studies on subtype discovery, network estimation, and epigenetic-wide association.