Aberrations in the 3D geometry of biological cells are linked to disease, and advances in microscopy have lead to rapid growth in 3D cell imaging. Despite this, there is a paucity of methods to quantify 3D cell shapes. Currently most descriptions of cell geometry use predefined mathematical measures, rather than data-driven approaches. To address this we have adapted existing geometric deep learning and improved deep embedded clustering (IDEC), and present a novel dynamic graph convolutional foldingnet autoencoder (DFN) with IDEC to simultaneously learn lower-dimensional representations and classes of 3D cell shapes from a dataset of more than 70,000 drug-treated melanoma cells imaged by 3D light-sheet microscopy. We propose to describe cell shape using 3D quantitative morphological signatures (3DQMS), representing a cell's similarity to shape modes in the dataset. This led to the insight that drugs treated with similar inhibitors share morphological signatures, which can be used to predict the activity of a drug. We also found that our model improves upon existing methods for problems such as classifying cell types based on geometry, by using a recently published dataset of 3D red blood cells. This suggests that our features generalise across datasets and that our geometric deep learning models are capturing features which are not explained by classical measures of shape. Finally, we highlight the implementation of our framework as a python software package for ease of use by the medical research community.