Unsupervised machine learning is widely used to mine large, unlabeled datasets to make data-driven discoveries in critical domains such as climate science, biomedicine, astronomy, chemistry, and more. However, despite its widespread utilization, there is a lack of standardization in unsupervised learning workflows for making reliable and reproducible scientific discoveries. In this paper, we present a structured workflow for using unsupervised learning techniques in science. We highlight and discuss best practices starting with formulating validatable scientific questions, conducting robust data preparation and exploration, using a range of modeling techniques, performing rigorous validation by evaluating the stability and generalizability of unsupervised learning conclusions, and promoting effective communication and documentation of results to ensure reproducible scientific discoveries. To illustrate our proposed workflow, we present a case study from astronomy, seeking to refine globular clusters of Milky Way stars based upon their chemical composition. Our case study highlights the importance of validation and illustrates how the benefits of a carefully-designed workflow for unsupervised learning can advance scientific discovery.