This paper intends to study the dispersion and attenuation characteristics of Love-type wave propagation due to a disturbance point source in a hydrostatic stressed magneto-viscoelastic layer, lying over a heterogeneous fibre-reinforced elastic half-space. The shear elastic moduli and mass density of half-space are the functions of depth and heterogeneity parameters. The coupled field equations are solved with the aid of Green's function technique and Fourier transform. The dispersion and damping equations have been obtained for the wave. The deduced equations coincide with the classical Love-wave condition for the uniform homogeneous isotropic structure. Numerical computations are carried out for involved parameters and demonstrated with the help of graphs. The effects of hydrostatic stress, magnetic coupling parameter, dissipation factor, attenuation coefficient, reinforcement parameters, heterogeneity parameters and order of the depth variation on the dispersion and damping curves are highlighted. © 2018 The Society of Theoretical and Applied Mechanics.