MicroRNAs (miRNAs) are known as an important indicator of cancers. The presence of cancer can be detected by identifying the responsible miRNAs. A fuzzy-rough entropy measure (FREM) is developed which can rank the miRNAs and thereby identify the relevant ones. FREM is used to determine the relevance of a miRNA in terms of separability between normal and cancer classes. While computing the FREM for a miRNA, fuzziness takes care of the overlapping between normal and cancer expressions, whereas rough lower approximation determines their class sizes. MiRNAs are sorted according to the highest relevance (i.e., the capability of class separation) and a percentage among them is selected from the top ranked ones. FREM is also used to determine the redundancy between two miRNAs and the redundant ones are removed from the selected set, as per the necessity. A histogram based patient selection method is also developed which can help to reduce the number of patients to be dealt during the computation of FREM, while compromising very little with the performance of the selected miRNAs for most of the data sets. The superiority of the FREM as compared to some existing methods is demonstrated extensively on six data sets in terms of sensitivity, specificity, and score. While for these data sets the score of the miRNAs selected by our method varies from 0.70 to 0.91 using SVM, those results vary from 0.37 to 0.90 for some other methods. Moreover, all the selected miRNAs corroborate with the findings of biological investigations or pathway analysis tools. The source code of FREM is available at http://www.jayanta.droppages.com/FREM.html.