As a novel fuzzy covering, fuzzy $\beta$ covering has attracted considerable attention. However, the traditional fuzzy-$\beta$-covering-based rough set and most of its extended models cannot well fit the distribution of samples in real data, which limits their application in classification learning and decision making. First, the upper and lower approximations of these models have no inclusion relation, so they cannot characterize a given objective concept accurately. Moreover, most of these models are hard to resist the influence of noise data, resulting in poor robustness in feature learning. For these reasons, a robust rough set model is set forth by combining fuzzy rough sets, covering-based rough sets, and multigranulation rough sets. To this end, the optimistic and pessimistic lower and upper approximations of a target concept are reconstructed by means of the fuzzy $\beta$ neighborhood related to a family of fuzzy coverings, and a new multigranulation fuzzy rough set model is presented. Furthermore, a fuzzy dependence function is introduced to evaluate the classification ability of a family of fuzzy $\beta$ coverings at different granularity levels. The dimensionality reduction of a given fuzzy covering decision table is carried out from the perspective of maintaining the discrimination power, and a forward algorithm for feature selection is developed by using the optimistic significance of candidate features as heuristic information. Three groups of numerical experiments on 16 different types of datasets demonstrate that the proposed model exhibits good robustness on datasets contaminated with noise and outperforms some state-of-the-art feature learning algorithms in terms of classification accuracy and the size of the selected feature subset.