Extracting geometric edges from unstructured point clouds remains a significant challenge, particularly in thin-walled structures that are commonly found in everyday objects. Traditional geometric methods and recent learning-based approaches frequently struggle with these structures, as both rely heavily on sufficient contextual information from local point neighborhoods. However, 3D measurement data of thin-walled structures often lack the accurate, dense, and regular neighborhood sampling required for reliable edge extraction, resulting in degraded performance.In this work, we introduce STAR-Edge, a novel approach designed for detecting and refining edge points in thin-walled structures. Our method leverages a unique representation—the local spherical curve—to create structure-aware neighborhoods that emphasize co-planar points while reducing interference from close-by, non-co-planar surfaces. This representation is transformed into a rotation-invariant descriptor, which, combined with a lightweight multi-layer perceptron, enables robust edge point classification even in the presence of noise and sparse or irregular sampling. Besides, we also use the local spherical curve representation to estimate more precise normals and introduce an optimization function to project initially identified edge points exactly on the true edges. Experiments conducted on the ABC dataset and thin-walled structure-specific datasets demonstrate that STAR-Edge outperforms existing edge detection methods, showcasing better robustness under various challenging conditions. The source code is available at https://github.com/miraclelzk/star-edge.