This article considers developing false discovery rate (FDR) controlling methods for testing multiple hypotheses under three different classification settings of the hypotheses into groups — simultaneous multi-way classification, hierarchical classification, and a combination of these two classifications. The methods are developed in their oracle forms by considering a weighted version of the Benjamini–Hochberg (BH, 1995) method, with the weights encoding the underlying structural information about the hypotheses. They control the FDR when the p-values involved are Positively Regression Dependent on the Subset (PRDS) of null p-values, and are more powerful than the BH procedure. Data-adaptive versions of these methods are also proposed by appropriately estimating the weights under the different types of classification. The proposed data-adaptive methods control the FDR at the desired level when the p-values are independent and, as simulations show, they can be more powerful FDR controlling methods, even under certain dependency, than some existing comparable multiple testing methods. We apply the data-adaptive method under the above-mentioned combined classification setting to analyze a publicly available neuro-imaging dataset. Such data typically have complex classification structures that were not addressed in previously available multiple testing methods, as far as we know. Our proposed method, with a flexible weighting scheme, is poised to utilize more information from the data in its decision making process, than other existing multiple testing methods.