The study proposes a fuzzy clustering algorithm referred to as the Sharma–Mittal divergence-regularized fuzzy c-means, which adopts the Sharma–Mittal divergence (SM-divergence) as the regularizer, where the SM-divergence is a generalization of the Tsallis and Rényi divergences. The algorithm is shown to reduce to two conventional algorithms, the Rényi and Tsallis divergence-based algorithms, by adjusting the parameter values at both the objective function and algorithmic levels. This study also examines the differences between the use of divergence in the Rényi and SM-divergence-based algorithms and object-wise divergences in the Kullback–Leibler and Tsallis divergence-based algorithms. Subsequently, this study proposed two fuzzy clustering algorithms by applying object-wise Rényi and SM-divergences. Numerical experiments on an artificial dataset verified the theoretical findings and highlighted the impact of the fuzzification parameter on clustering results.