DNA replication occurs throughout the S phase of the cell cycle, initiating from replication origin loci that fire at different times. Debate remains about whether origins are a fixed set of loci used across all cells or a loose agglomeration of potential origins used stochastically in individual cells, and about how consistent their firing time during S phase is across cells. Here, we develop an approach for profiling DNA replication in single human cells and apply it to 2,305 replicating cells spanning the entire S phase. The resolution and scale of the data enabled us to specifically analyze initiation sites and show that these sites have confined locations that are consistently used among individual cells. Further, we find that initiation sites are activated in a similar, albeit not fixed, order across cells. Taken together, our results suggest that replication timing variability is constrained both spatially and temporally, and that the degree of variation is consistent across human cell lines.