Three-dimensional asymptotic solutions are obtained for magneto-electro-elastic singularities in bodies of revolution that are made of a single magneto-electro-elastic (MEE) material or bi-materials that consist of an MEE material and an elastic material or piezoelectric material. The solutions are obtained by combining an eigenfunction expansion approach with the power series solution method to solve three-dimensional equilibrium equations and Maxwell's equations in terms of mechanical displacement components and electric and magnetic potentials. The MEE material is assumed to be transversely isotropic and its polarization direction is not necessarily parallel to the axis of revolution. The polarization direction, which is not along the axis of revolution, yields and complicates non-axisymmetric solutions. The solutions are validated by comparing the present characteristic values of the asymptotic solutions, which are related to the orders of singularities of stress, electric displacement and magnetic flux, to the published ones for a piezoelectric body of revolution because no results have been published for an MEE body of revolution. The developed solutions are further employed to examine the effects of the direction of polarization, the configuration of the body of revolution and the material components on the orders of the singularities in bodies of revolution that comprise a single MEE material (BaTiO3-CoFe2O4) and bonded MEE/isotropic elastic (BaTiO3-CoFe2O4/Si), MEE/piezoelectric (BaTiO3-CoFe2O4/PZT-5H), or MEE/MEE BaTiO3-CoFe2O4VI=50%/BaTiO3-CoFe2O4VI=20% materials. These results are published here for the first time.