This paper proposes a new scheme for detecting ship targets in high-resolution (HR) single-channel synthetic aperture radar (SAR) images. By using the proposed spatially enhanced pixel descriptor (SEPD) and the modified density-based spatial clustering of application with noise (M-DBSCAN), this scheme can overcome typical challenges of ship detection in HR SAR images. Specifically, the proposed SEPD maps the representation for a given pixel in an SAR image into a high-dimensional feature space by embedding spatial and intensity information of its neighborhood synchronously. It enables the spatial structure information of ship targets and textural information of the sea surface to be preserved by the SEPD feature vector, leading to a significant improvement in the separability between ship targets and sea clutter. A statistical study shows that, in SEPD feature space, a large amount of pixels belonging to sea clutter gather densely in the low-value region and are surrounded by a tiny proportion of ship targets that are distributed sparsely in the high-value region. This distribution characteristic motivates us to apply a density-based clustering approach to distinguish ship targets from the sea clutter. To overcome the weakness of original DBSCAN clustering algorithm and make it suitable for the requirements of ship detection in SAR images, we propose the method of M-DBSCAN, which introduces three critical improvements, including a new dimensionality independent distance metric, a one-class clustering strategy, and an entirely deterministic approach to border points. A novel ship detector is proposed by applying M-DBSCAN to cluster pixels that are represented by SEPD descriptor. Comprehensive experiments demonstrate that the proposed method outperforms other intensity-based clustering methods (k-means and fuzzy c-means), and widely used intensity threshold-based method (constant false alarm rate detector) in most circumstances, and can effectively handle various challenging situations appearing in HR images, such as sidelobes, small/ weak targets, moving targets, and so on.