Feature Selection (FS) is a challenging machine learning-related task that aims at reducing the number of features by removing irrelevant, redundant and noisy data while maintaining an acceptable level of classification accuracy. FS can be considered as an optimisation problem. Due to the difficulty of this problem and having a large number of local solutions, stochastic optimisation algorithms are promising techniques to solve this problem. As a seminal attempt, binary variants of the recent Grasshopper Optimisation Algorithm (GOA) are proposed in this work and employed to select the optimal feature subset for classification purposes within a wrapper-based framework. Two mechanisms are employed to design a binary GOA, the first one is based on Sigmoid and V-shaped transfer functions, and will be indicated by BGOA-S and BGOA-V, respectively. While the second mechanism uses a novel technique that combines the best solution obtained so far. In addition, a mutation operator is employed to enhance the exploration phase in BGOA algorithm (BGOA-M). The proposed methods are evaluated using 25 standard UCI datasets and compared with 8 well-regarded metaheuristic wrapper-based approaches, and six well known filter-based (e.g., correlation FS) approaches. The comparative results show the superior performance of the BGOA and BGOA-M methods compared to other similar techniques in the literature.