Modeling cell shape variation is critical to our understanding of cell biology. Previous work has demonstrated the utility of non-rigid image registration methods for the construction of non-parametric nuclear shape models where pairwise deformation distances are measured between all shapes and are embedded into a low dimensional shape space. Using these methods we explore the relationship between cell shape and nuclear shape. We find that these are frequently dependent upon each other and use this as the motivation for the development of combined cell and nuclear shape space models, extending non-parametric cell representations to multiple component 3D cellular shapes and identifying modes of joint shape variation. We learn a first-order dynamics model to predict cell and nuclear shapes given shapes at a previous time point. We use this to determine the effects of endogenous protein tags or drugs on the shape dynamics of cell lines, and show that tagged C1QBP reduces the correlation between cell and nuclear shape. To reduce the computational cost of learning these models, we demonstrate the ability to reconstruct shape spaces using a fraction of computed pairwise distances. The open source tools provide a powerful basis for future studies of the molecular basis of cell organization.