Two-dimensional (2D) and three-dimensional (3D) Cun 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. [ABSTRACT FROM AUTHOR]