The time evolution of the Hamming distance (damage spreading) for the S = 1/2 and S = 1 Ising models on the square lattice is performed with a special metropolis dynamics algorithm. Two distinct regimes are observed according to the temperature range for both models: a low-temperature one where the distance in the long-time limit is finite and seems not to depend on the initial distance and the system size; a high-temperature one where the distance vanishes in the long-time limit. Using the finite size scaling method, the dynamical phase transition (damage spreading transition) temperature is obtained as Tc ≌ 1.675 ± 0.025 for the S = 1 Ising model.