Generic sentence embeddings provide coarse-grained approximation of semantic textual similarity, but ignore specific aspects that make texts similar. Conversely, aspect-based sentence embeddings provide similarities between texts based on certain predefined aspects. Thus, similarity predictions of texts are more targeted to specific requirements and more easily explainable. In this paper, we present AspectCSE, an approach for aspect-based contrastive learning of sentence embeddings. Results indicate that AspectCSE achieves an average improvement of 3.97% on information retrieval tasks across multiple aspects compared to the previous best results. We also propose the use of Wikidata knowledge graph properties to train models of multi-aspect sentence embeddings in which multiple specific aspects are simultaneously considered during similarity predictions. We demonstrate that multi-aspect embeddings outperform even single-aspect embeddings on aspect-specific information retrieval tasks. Finally, we examine the aspect-based sentence embedding space and demonstrate that embeddings of semantically similar aspect labels are often close, even without explicit similarity training between different aspect labels.