The current study examines the behavior of an SH wave traveling over a functionally graded magneto-elastic substrate arrangement. At the substrate-vacuum interface, two irregularities with different shapes-rectangular and parabolically shaped-are considered in electrically and magnetically open cases and electrically and magnetically short cases. A study is also done on the combined impact of inhomogeneity, depth source, and irregularity. With the help of the Fourier transform, inverse Fourier transform, and perturbation technique, complex frequency relation has been derived for each type of irregular interface. The results' key characteristics are highlighted. In order to know the impact of the parameters involved, a particular model consisting of BaTiO3-CoFe2O4 magneto-electro-elastic material has been taken. The findings were presented in the form of graphs, which were created using Mathematica 7. Graphs are plotted for variations in wavenumber and phase velocity. This calculation model could be the ideal match for laminated FGMEE structures utilized as surface acoustic wave devices since the variation of the film's magneto-electromechanical characteristics changes gradually with depth and throughout the production process (SAW). As a result, it can serve as a theoretical foundation for the design of high-performance SAW devices.