Persistence diagrams concisely capture the structure of data, an ability that is increasingly being used in the nascent field of topological machine learning. We extend the ubiquitous Fuzzy c-Means (FCM) clustering algorithm to the space of persistence diagrams, enabling unsupervised learning in a topological setting. We give theoretical convergence guarantees that correspond to the Euclidean case and empirically demonstrate the capability of the clustering to capture topological information via the fuzzy RAND index. We present an application of our algorithm to a scenario that utilises both the topological and fuzzy nature of our algorithm: pre-trained model selection in deep learning. As pre-trained models can perform well on multiple tasks, selecting the best model is a naturally fuzzy problem; we show that fuzzy clustering persistence diagrams allows for unsupervised model selection using just the topology of their decision boundaries.