Haptic perception remains a grand challenge for artificial hands. Dexterous manipulators could be enhanced by “haptic intelligence” that enables identification of objects and their features via touch alone. Haptic perception of local shape would be useful when vision is obstructed or when proprioceptive feedback is inadequate, as observed in this study. In this work, a robot hand outfitted with a deformable, bladder-type, multimodal tactile sensor was used to replay four human-inspired haptic “exploratory procedures” on fingertip-sized geometric features. The geometric features varied by type (bump, pit), curvature (planar, conical, spherical), and footprint dimension (1.25 - 20 mm). Tactile signals generated by active fingertip motions were used to extract key parameters for use as inputs to supervised learning models. A support vector classifier estimated order of curvature while support vector regression models estimated footprint dimension once curvature had been estimated. A distal-proximal stroke (along the long axis of the finger) enabled estimation of order of curvature with an accuracy of 97%. Best-performing, curvature-specific, support vector regression models yielded R2 values of at least 0.95. While a radial-ulnar stroke (along the short axis of the finger) was most helpful for estimating feature type and size for planar features, a rolling motion was most helpful for conical and spherical features. The ability to haptically perceive local shape could be used to advance robot autonomy and provide haptic feedback to human teleoperators of devices ranging from bomb defusal robots to neuroprostheses.