Mammalian interphase chromosomes interact with the nuclear lamina (NL) through hundreds of large lamina-associated domains (LADs). We report a method to map NL contacts genome-wide in single human cells. Analysis of nearly 400 maps reveals a core architecture consisting of gene-poor LADs that contact the NL with high cell-to-cell consistency, interspersed by LADs with more variable NL interactions. The variable contacts tend to be cell-type specific and are more sensitive to changes in genome ploidy than the consistent contacts. Single-cell maps indicate that NL contacts involve multivalent interactions over hundreds of kilobases. Moreover, we observe extensive intra-chromosomal coordination of NL contacts, even over tens of megabases. Such coordinated loci exhibit preferential interactions as detected by Hi-C. Finally, the consistency of NL contacts is inversely linked to gene activity in single cells and correlates positively with the heterochromatic histone modification H3K9me3. These results highlight fundamental principles of single-cell chromatin organization.