The C-H breaking of methane on the clean and the oxygen precovered palladium single crystal surfaces with the simplest orientations, namely, the dense (111), (100), the more open (110), and the stepped (111) surfaces, the corresponding O/Pd surfaces with different coverage of oxygen, as well as the palladium oxide PdO(100) and PdO(110) surfaces, has been studied with the density functional theory-generalized gradient approximation method using the repeated slab models. The adsorption energies under the most stable configuration of the possible species and the activation energy barriers of the reaction are obtained in the present work. Through systematic calculations for the C-H breaking of methane CH(4)-->CH(3)+H on these surfaces, it is found that such a reaction is structure sensitive on clean palladium and oxygen precovered palladium surfaces with lower oxygen coverage, but it is insensitive on oxygen precovered palladium surfaces with higher oxygen coverage and on palladium oxides. These results are in general agreement with the experimental observations.