The rate constant (W) of catalytic dissociation of a methane molecule to adsorbed CH3 and H and its temperature dependence are studied in the framework of a tunneling model which considers the reaction as a result of quantum transitions between discrete vibrational energy levels of an initial C-H bond in methane and energy continuum of the dissociated final state while the system reaches the transition state along other classical degrees of freedom. This step is assumed to be the rate determining one in many reactions of commercial significance like steam and dry reforming of methane. This approach has previously been successfully applied to other catalytic reactions of H atom transfer. Numerical calculations of the kinetic characteristics for the reaction proceeding over surfaces of Ni(111), Pt(111), Cu(111), Pd(111), Ru(001), and Ir(111) are interpreted in terms of the Brønsted-Evans-Polanyi (BEP) relationship. The good correlation and the theory show that the main discriminating parameter which determines the trend within a set of metals is the total adsorption energy of a methyl fragment and an H atom. The BEP relationship is used for prediction of the rate constants of the dissociation on rhodium, silver, and gold (111) surfaces and for predicting the activation energy on bimetallic surfaces. Sticking coefficients are estimated using the W values and adsorption energy of the methane molecule; the latter is approximately assumed to be the same for all considered surfaces. These estimations are compared with the experimental sticking coefficients for several metals showing quantitative agreement. Hydrogen kinetic isotope effects (KIEs) and their temperature dependencies are calculated for the dissociation on nickel and ruthenium surfaces, and the KIE estimates at fixed temperatures of iridium and rhodium are compared with the corresponding experimental data. Results of this comparison justify the application of the model for the description of the catalytic dissociation of the C-H bond.