The spatial organization of the genome is essential for its functions, including gene expression and chromosome segregation. Phase separation and loop extrusion have been proposed to underlie compartments and topologically associating domains, however, whether the fold of genomic DNA inside the nucleus is consistent with such mechanisms has been difficult to establish in situ. Here, we present a 3D DNA-tracing workflow that resolves genome architecture in single structurally well-preserved cells with nanometre resolution. Our findings reveal that genomic DNA generally behaves as a flexible random coil at the 100-kb scale. At CTCF sites however, we find Cohesin-dependent loops in a subset of cells, in variable conformations from the kilobase to megabase scale. The 3D-folds we measured in hundreds of single cells allowed us to formulate a computational model that explains how sparse and dynamic loops in single cells underlie the appearance of compact topological domains measured in cell populations.