We present single-cell combinatorial indexed Hi-C (sciHi-C), a method that applies combinatorial cellular indexing to chromosome conformation capture. In this proof of concept, we generate and sequence six sciHi-C libraries comprising a total of 10,696 single cells. We use sciHi-C data to separate cells by karyotypic and cell-cycle state differences and identify cell-to-cell heterogeneity in mammalian chromosomal conformation. Our results demonstrate that combinatorial indexing is a generalizable strategy for single-cell genomics.