The unique optical and physical properties of surface plasmon polaritons (SPP) has brought about a series of novel phenomena such as SPP-enhanced transmission, local resonance, etc., and SPP has become a research hotspot around the world. In this paper, the dispersion characteristics and modes of rectangular metal grating based on spoof surface plasmons (SSP) are studied theoretically and numerically. The electromagnetic fields of SSP which are below and above the grating surface are presented using eigenmode expansion method and under periodic boundary conditions, besides the fact that the SSP dispersion relations are obtained by matching the boundary conditions of electromagnetic fields both for rectangular metal grating with roofed metal plate and that without roofed metal plate. Results for these two different cases are given according to numerical calculation and it is found that the roofed metal plate can introduce an additional fast wave mode which is beyond the light line in the dispersion diagram. And the results of analytical SSP dispersion are verified by electromagnetic simulations based on the finite difference method and finite integration method. The dependence of the dispersion characteristics and mode distributions on various parameters of metal grating is studied theoretically. It is shown that the dispersion relations obtained by eigenmode expansion method agree well with the results of electromagnetic simulations. The phase velocity of SSP on the grating surface can be decreased by increasing metal grating depth or decreasing grating period. The bandwidth of electron beam-SSP interaction can be extended by increasing grating period ratio. The influence of the distance between the roofed metal plate and the grating surface on the SSP dispersion is studied and is found that the role of roofed metal plate is insensitive to the slow wave SSP mode. The SSP dispersion and modes for the 3-D metal grating which are extended from the above 2-D SSP dispersion are also given. The SSP symmetric modes and anti-symmetric modes manifest themself alternately in the dispersion diagram on the 3-D grating surface. Compared with the 2-D SSP bound mode without roofed metal plate, it is found that in the 3-D grating structure the slow wave SSP modes and fast wave SSP modes coexist. The 3-D SSP mode with various grating lateral width is studied, and the competition and degeneracy of modes are analyzed particularly. The SSP mode intervals can be enlarged by decreasing the lateral width of the grating, which is optimum for avoiding mode competitions. Studies on dispersion and modes of the 2-D and 3-D metal grating structures based on SSP will lay the foundations for further studies of electron beam-SSP interaction, and development of the novel terahertz vacuum electronic source with high-efficiency and wide-bandwidth.