Two-dimensional (2D) and three-dimensional (3D) Cu n clusters (n indicates the atom number) and their adsorption behaviors for both methane (CH4) and methyl (CH3) are studied in this work using the density functional theory method, where n ranges from 6 to 20. In these small clusters, it is found that the CH4 molecule is always adsorbed on the top site with the adsorption energy between -0.05 eV and -0.21 eV. Considering methane dehydrogenation, stronger adsorption for CH4 is required, so 2D clusters with n = 7, 14, 15, and 16 and 3D clusters with n = 6, 10, 12, and 17 are found to have relatively stronger adsorption. However, for the adsorption of CH3, there is an obvious even-odd oscillation change in the size of 3D clusters, while it is not clear in 2D clusters since one cannot find an even-odd change as n > 14. The weaker adsorption for CH3 occurs on 3D clusters when n is even except 6 and also on 2D clusters when n = 6, 7, 10, and 12 with higher carbon poisoning resistance. Based on these calculated results, some Cu clusters which show good potential ability for methane dehydrogenation are provided, especially when n = 10 and 12 for 3D structures, and n = 7 for the 2D ones.