Although a number of advances have been made in RNA sequencing and structural characterization, the lack of a method for directly determining the sequence and structure of single RNA molecules has limited our ability to probe heterogeneity in gene expression at the level of single cells. Here we present a method for direct nucleotide identification and structural label mapping of single RNA molecules via Quantum Molecular Sequencing (QMSeq). The method combines non-perturbative quantum tunneling spectroscopy to probe the molecular orbitals of ribonucleotides, new experimental biophysical parameters that fingerprint these molecular orbitals, and a machine learning classification algorithm to distinguish between the ribonucleotides. The algorithm uses tunneling spectroscopy measurements on an unknown ribonucleotide to determine its chemical identity and the presence of local chemical modifications. Combining this with structure-dependent chemical labeling presents the possibility of mapping both the sequence and local structure of individual RNA molecules. By optimizing the base-calling algorithm, we show a high accuracy for both ribonucleotide discrimination (>99.8%) and chemical label identification (>98%) with a relatively modest molecular coverage (35 repeat measurements). This lays the groundwork for simultaneous sequencing and structural mapping of single unknown RNA molecules, and paves the way for probing the sequence-structure-function relationship within the transcriptome at an unprecedented level of detail.