In this paper we consider the problem of classifying shapes within a given category (e.g., chairs) into finer-grained classes (e.g., chairs with arms, rocking chairs, swivel chairs). We introduce a multi-label (i.e., shapes can belong to multiple classes) semi-supervised approach that takes as input a large shape collection of a given category with associated sparse and noisy labels, and outputs cleaned and complete labels for each shape. The key idea of the proposed approach is to jointly learn a distance metric for each class which captures the underlying geometric similarity within that class, e.g., the distance metric for swivel chairs evaluates the global geometric resemblance of chair bases. We show how to achieve this objective by first geometrically aligning the input shapes, and then learning the class-specific distance metrics exploiting the feature consistency provided by this alignment. The learning objectives consider both labeled data and the mutual relations between the distance metrics. Given the learned metrics, we apply a graph-based semi-supervised classification technique to generate the final classification results. In order to evaluate the performance of the approach, we have created a benchmark data set where each shape is provided with a set of ground truth labels generated by Amazon's Mechanical Turk users. The benchmark contains a rich variety of shapes in a number of categories. Experimental results show that despite this variety, given very sparse and noisy initial labels, the new method yields results that are superior to state-of-the-art semi-supervised learning techniques.