Somatic structural variants (SVs) are widespread in cancer, but their impact on disease evolution is understudied due to a lack of methods to directly characterize their functional consequences. We present a computational method, scNOVA, which uses Strand-seq to perform haplotype-aware integration of SV discovery and molecular phenotyping in single cells by using nucleosome occupancy to infer gene expression as a readout. Application to leukemias and cell lines identifies local effects of copy-balanced rearrangements on gene deregulation, and consequences of SVs on aberrant signaling pathways in subclones. We discovered distinct SV subclones with dysregulated Wnt signaling in a chronic lymphocytic leukemia patient. We further uncovered the consequences of subclonal chromothripsis in T cell acute lymphoblastic leukemia, which revealed c-Myb activation, enrichment of a primitive cell state and informed successful targeting of the subclone in cell culture, using a Notch inhibitor. By directly linking SVs to their functional effects, scNOVA enables systematic single-cell multiomic studies of structural variation in heterogeneous cell populations.