Precise understanding of free-text radiology reports through localised extraction of clinical findings can enhance medical imaging applications like computer-aided diagnosis. We present a new task, that of segmenting radiology reports into topically meaningful passages (segments) and a transformer-based model that both segments reports into semantically coherent segments and classifies each segment using a set of 37 radiological abnormalities, thus enabling fine-grained analysis. This contrasts with prior work that performs classification on full reports without localisation. Trained on over 2.7 million unlabelled chest X-ray reports and over 28k segmented and labelled reports, our model achieves state-of-the-art performance on report segmentation (0.0442 WinDiff) and multi-label classification (0.84 report-level macro F1) over 37 radiological labels and 8 NLP-specific labels. This work establishes new benchmarks for fine-grained understanding of free-text radiology reports, with precise localisation of semantics unlocking new opportunities to improve computer vision model training and clinical decision support. We open-source our annotation tool, model code and pretrained weights to encourage future research.