The problem of computer-aided classification of benign and malignant breast masses using shape features is addressed. The aim of the study is to look at the exceptions in shapes of masses such as circumscribed malignant tumours and spiculated benign masses which are difficult to classify correctly using common shape analysis methods. The proposed methods of shape analysis treat the object's boundary in terms of local details. The boundaries of masses analysed using the proposed methods were manually drawn on mammographic images by an expert radiologist (JELD). A boundary segmentation method is used to separate major portions of the boundary and to label them as concave or convex segments. To analyse the shape information localised in each segment, features are computed through an iterative procedure for polygonal modelling of the mass boundaries. Features are based on the concavity fraction of a mass boundary and the degree of narrowness of spicules as characterised by a spiculation index. Two features comprising spiculation index (SI) and fractional concavity (fcc) developed in the present study when used in combination with the global shape feature of compactness resulted in a benign/malignant classification accuracy of 82%, with an area (Az) of 0.79 under the receiver operating characteristics (ROC) curve with a database of the boundaries of 28 benign masses and 26 malignant tumours. SI alone resulted in a classification accuracy of 80% with Az of 0.82. The combination of all the three features achieved 91% accuracy of circumscribed versus spiculated classification of masses based on shape.