Skill discovery has emerged as a popular route for unsupervised reinforcement learning (URL), offering agents a diverse, reusable set of behaviours learned before any task-specific reward is experienced. However, existing methodologies tend to favour either categorical codes or unimodal skill priors, which simplifies training at the cost of limiting the variety of behaviours they can represent. We introduce \emph{Discovery of Mixture Skills} (DiMS), a URL algorithm that learns a latent Gaussian mixture by training a Gaussian Mixture Variational Autoencoder (GMVAE) in tandem with the unsupervised policy. In DiMS, a hierarchical GMVAE simultaneously discovers clusters of skills, while an auxiliary macro-latent dynamically positions mixture components to prevent mode collapse. A joint loss term combining log-likelihood and curiosity rewards enables simultaneous updates of representation and policy while improving exploration. Experiments on the Unsupervised Reinforcement Learning Benchmark (URLB) show that DiMS consistently outperforms a wide range of state-of-the-art baselines. Ablation studies confirm that the mixture prior is critical to these gains, and that DiMS is robust to alternative exploration bonuses. Overall, our results suggest that Gaussian mixture skill priors offer a compelling foundation for future unsupervised RL.