© 2022, The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.In this work, we aim to derive various soliton solutions to the Wazwaz–Benjamin–Bona–Mahony equation with conformable and M-truncated derivatives. The considered equation models long waves in the ocean engineering field. Unified Riccati equation expansion and Kudryashov auxiliary equation methods are used to the model, and so, kink, singular, and periodic-singular soliton solutions are successfully obtained. It is also reported on the constraint conditions that assure the validity of novel wave forms. By choosing appropriate parameters, numerical simulations of the obtained results are depicted by using two- and three-dimensional plots and the comparative results between the solutions for the conformable and M-truncated derivative are shown in two-dimensional graphs for various orders α, β. Moreover, the effects of the parameters in the obtained solutions are shown. The methods might be useful for obtaining the analytical solutions of many physical phenomena in nature since they are effective, robust, and easily applicable. Finally, this study contributes to extract both various solutions to the literature and to investigate wave behavior while the parameters change.