The present paper is devoted to creation of an efficient strategy for finding the optimum values of various hyperparameters characterizing spiking neural networks (SNN) and models of spiking neurons constituting them. This problem is also important for traditional artificial neural networks (ANN). However, discrete and stochastic nature of SNNs is a great obstacle for application of the elaborated numeric optimization methods efficient in the “smoother” domain of ANN. The problem is aggravated by insufficient development of theoretical methods for SNN exploration and difficulties of SNN simulation on usual computers. It forced us to use the case study approach – we selected one particular SNN architecture, namely CoLaNET (an SNN solving classification problems), and one particular task (MNIST classification). For this case, we explored thoroughly the hyperparameter optimization methodology based on combination of genetic algorithm and stochastic descent, the stable and general methods suitable for very “humpy” optimization surfaces. On the basis of this study, we have worked out the SNN hyperparameter optimization protocol minimizing the total number of simulations necessary to reach acceptable network performance. While it is impossible to prove that this protocol is the best for all SNN hyperparameter optimization problems, our choice of the typical image classification task and the general-purpose SNN architecture for this case study enables us to consider it, at least, as a good starting point for any SNN optimization task.