Influenced by the complexity of volumetric imaging, there is a shortage of established datasets useful for benchmarking volumetric deep-learning models. As a consequence, new and existing models are not easily comparable, limiting the development of architectures optimized specifically for volumetric data. To counteract this trend, we introduce MozzaVID -- a large, clean, and versatile volumetric classification dataset. Our dataset contains X-ray computed tomography (CT) images of mozzarella microstructure and enables the classification of 25 cheese types and 149 cheese samples. We provide data in three different resolutions, resulting in three dataset instances containing from 591 to 37,824 images. While targeted for developing general purpose volumetric algorithms, the dataset also facilitates investigating the properties of mozzarella microstructure. The complex and disordered nature of food structures brings a unique challenge, where a choice of appropriate imaging method, scale, and sample size is not trivial. With this dataset, we aim to address these complexities, contributing to more robust structural analysis models and a deeper understanding of food structure.