Our understanding of the beads-on-a-string arrangement of nucleosomes has been built largely on high-resolution sequence-agnostic imaging methods and sequence-resolved bulk biochemical techniques. To bridge the divide between these approaches, we present the single-molecule adenine methylated oligonucleosome sequencing assay (SAMOSA). SAMOSA is a high-throughput single-molecule sequencing method that combines adenine methyltransferase footprinting and single-molecule real-time DNA sequencing to natively and nondestructively measure nucleosome positions on individual chromatin fibres. SAMOSA data allows unbiased classification of single-molecular 'states' of nucleosome occupancy on individual chromatin fibres. We leverage this to estimate nucleosome regularity and spacing on single chromatin fibres genome-wide, at predicted transcription factor binding motifs, and across human epigenomic domains. Our analyses suggest that chromatin is comprised of both regular and irregular single-molecular oligonucleosome patterns that differ subtly in their relative abundance across epigenomic domains. This irregularity is particularly striking in constitutive heterochromatin, which has typically been viewed as a conformationally static entity. Our proof-of-concept study provides a powerful new methodology for studying nucleosome organization at a previously intractable resolution and offers up new avenues for modeling and visualizing higher order chromatin structure.