The clustering of data points in n-dimensional Euclidean space, i. e., assigning each data point to exactly one group (cluster) to detect previously unseen relations within the data set, has become a standard task for (unsupervised) machine learning. In this paper, this concept is generalized to shape data consisting of three-dimensional volumetric objects. As the underlying distance on the space of such objects, the optimal transport-based Wasserstein distance is considered and different variants of such a clustering approach are presented and compared. A variational autoencoder representing these cluster centers is incorporated to counteract an over-smoothing of objects representing the cluster centers. Numerical experiments for three distinct, volumetric data sets are presented to validate the performance of the proposed methods.