© 2022This paper extracts some analytical solutions of simplified modified Camassa-Holm (SMCH) equations with various derivative operators, namely conformable and M-truncated derivatives that have been recently introduced. The SMCH equation is used to model the unidirectional propagation of shallow-water waves. The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived. We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system. Furthermore, some 2D and 3D graphical illustrations of the obtained solutions have been presented. The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted. The derived solutions may contribute to comprehending wave propagation in shallow water. So, the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management, which are critical topics in the ocean and coastal engineering.