An emerging trend in feature selection is the development of two-objective algorithms that analyze the tradeoff between the number of features and the classification performance of the model built with these features. Since these two objectives are conflicting, a typical result stands in a set of Pareto-efficient subsets, each having a different cardinality and a corresponding discriminating power. However, this approach overlooks the fact that, for a given cardinality, there can be several subsets with similar information content. The study reported here addresses this problem, and introduces a novel multiobjective feature selection approach conceived to identify: 1) a subset that maximizes the performance of a given classifier and 2) a set of subsets that are quasi equally informative, i.e., have almost same classification performance, to the performance maximizing subset. The approach consists of a wrapper [Wrapper for Quasi Equally Informative Subset Selection (W-QEISS)] built on the formulation of a four-objective optimization problem, which is aimed at maximizing the accuracy of a classifier, minimizing the number of features, and optimizing two entropy-based measures of relevance and redundancy. This allows conducting the search in a larger space, thus enabling the wrapper to generate a large number of Pareto-efficient solutions. The algorithm is compared against the mRMR algorithm, a two-objective wrapper and a computationally efficient filter [Filter for Quasi Equally Informative Subset Selection (F-QEISS)] on 24 University of California, Irvine, (UCI) datasets including both binary and multiclass classification. Experimental results show that W-QEISS has the capability of evolving a rich and diverse set of Pareto-efficient solutions, and that their availability helps in: 1) studying the tradeoff between multiple measures of classification performance and 2) understanding the relative importance of each feature. The quasi equally informative subsets are identified at the cost of a marginal increase in the computational time thanks to the adoption of Borg Multiobjective Evolutionary Algorithm and Extreme Learning Machine as global optimization and learning algorithms, respectively.