Epigenetic control of gene expression is highly cell-type- and context-specific. Yet, despite its complexity, gene regulatory logic can be broken down into modular components consisting of a transcription factor (TF) activating or repressing the expression of a target gene through its binding to a cis-regulatory region. Recent advances in joint profiling of transcription and chromatin accessibility with single-cell resolution offer unprecedented opportunities to interrogate such regulatory logic. Here, we propose a nonparametric approach, TRIPOD, to detect and characterize three-way relationships between a TF, its target gene, and the accessibility of the TF’s binding site, using single-cell RNA and ATAC multiomic data. We apply TRIPOD to interrogate cell-type-specific regulatory logic in peripheral blood mononuclear cells and contrast our results to detections from enhancer databases, cis-eQTL studies, ChIP-seq experiments, and TF knockdown/knockout studies. We then apply TRIPOD to mouse embryonic brain data during neurogenesis and gliogenesis and identified known and novel putative regulatory relationships, validated by ChIP-seq and PLAC-seq. Finally, we demonstrate TRIPOD on SHARE-seq data of differentiating mouse hair follicle cells and identify lineage-specific regulation supported by histone marks for gene activation and super-enhancer annotations.