A novel fault diagnosis scheme was developed to address the difficulty of feature extraction for planetary gearboxes using refined composite multiscale fluctuation dispersion entropy (RCMFDE) and supervised manifold mapping. The RCMFDE was first utilized in this scheme to fully mine fault features from planetary gearbox signals under multiple scales. Subsequently, as a supervised manifold mapping method, supervised isometric mapping (S-Iso) was applied to decrease the dimensions of the original features and remove redundant information. Lastly, the marine predator algorithm-based support vector machine (MPA-SVM) classifier was employed to achieve intelligent fault diagnosis of planetary gearboxes. The suggested RCMFDE combines the composite coarse-grained construction and refined computing technology, overcoming unstable and invalid entropy in the traditional multiscale fluctuation dispersion entropy. Simulation experiments and fault diagnosis experiments from a real planetary gearbox drive system show that the complexity measure capability and feature extraction effectiveness of the proposed RCMFDE outperform the multiscale fluctuation dispersion entropy (MFDE) and multi-scale permutation entropy (MPE). The S-Iso’s visualization results and dimensionality reduction performance are better than principal components analysis (PCA), linear discriminant analysis (LDA), and isometric mapping (Isomap). Moreover, the suggested fault diagnosis scheme has an accuracy rate of 100% in identifying bearing and gear defects in planetary gearboxes.