  With Monte Carlo methods, we investigate the universality class of the
depinning transition in the two-dimensional Ising model with quenched random
fields. Based on the short-time dynamic approach, we accurately determine the
depinning transition field and both static and dynamic critical exponents. The
critical exponents vary significantly with the form and strength of the random
fields, but exhibit independence on the updating schemes of the Monte Carlo
algorithm. From the roughness exponents $\zeta, \zeta_{loc}$ and $\zeta_s$, one
may judge that the depinning transition of the random-field Ising model belongs
to the new dynamic universality class with $\zeta \neq \zeta_{loc}\neq \zeta_s$
and $\zeta_{loc} \neq 1$. The crossover from the second-order phase transition
to the first-order one is observed for the uniform distribution of the random
fields, but it is not present for the Gaussian distribution.
