Chromatin folding contributes to the regulation of genomic processes such as gene activity. Existing conformation capture methods characterize genome topology through analysis of pairwise chromatin contacts in populations of cells but cannot discern whether individual interactions occur simultaneously or competitively. Here we present multi-contact 4C (MC-4C), which applies Nanopore sequencing to study multi-way DNA conformations of individual alleles. MC-4C distinguishes cooperative from random and competing interactions and identifies previously missed structures in subpopulations of cells. We show that individual elements of the β-globin superenhancer can aggregate into an enhancer hub that can simultaneously accommodate two genes. Neighboring chromatin domain loops can form rosette-like structures through collision of their CTCF-bound anchors, as seen most prominently in cells lacking the cohesin-unloading factor WAPL. Here, massive collision of CTCF-anchored chromatin loops is believed to reflect 'cohesin traffic jams'. Single-allele topology studies thus help us understand the mechanisms underlying genome folding and functioning.