Cluster algorithms are increasingly popular in biomedical research due to their compelling ability to identify discrete subgroups in data, and their increasing accessibility in mainstream software. While guidelines exist for algorithm selection and outcome evaluation, there are no firmly established ways of computing a priori statistical power for cluster analysis. Here, we estimated power and accuracy for common analysis pipelines through simulation. We varied subgroup size, number, separation (effect size), and covariance structure. We then subjected generated datasets to dimensionality reduction (none, multidimensional scaling, or UMAP) and cluster algorithms (k-means, agglomerative hierarchical clustering with Ward or average linkage and Euclidean or cosine distance, HDBSCAN). Finally, we compared the statistical power of discrete (k-means), "fuzzy" (c-means), and finite mixture modelling approaches (which include latent profile and latent class analysis). We found that outcomes were driven by large effect sizes or the accumulation of many smaller effects across features, and were unaffected by differences in covariance structure. Sufficient statistical power was achieved with relatively small samples (N=20 per subgroup), provided cluster separation is large (Δ=4). Fuzzy clustering provided a more parsimonious and powerful alternative for identifying separable multivariate normal distributions, particularly those with slightly lower centroid separation (Δ=3). Overall, we recommend that researchers 1) only apply cluster analysis when large subgroup separation is expected, 2) aim for sample sizes of N=20 to N=30 per expected subgroup, 3) use multidimensional scaling to improve cluster separation, and 4) use fuzzy clustering or finite mixture modelling approaches that are more powerful and more parsimonious with partially overlapping multivariate normal distributions.