For a class of nonlinear systems disturbed by multiple random noises and multiple random impulses, where the multiple random noises in continuous dynamics are composed of multiplicative and additive noises and the kind of multiple random impulsive amplitudes in discrete dynamics are driven by the Markov chain, this paper proposes the criteria of the noise-to-state stability in probability and in m-th moment, the global asymptotic stability in probability and the exponential stability in m-th moment, respectively. When the impulsive number is constrained by the mode-dependent average impulsive interval, firstly, the noise-to-state stability and exponential stability in m-th moment are investigated based on the estimation of multiplicative random noise, respectively. Then, on the basis of the assumption that the multiplicative random noise satisfies the law of large numbers, the sufficient conditions of the noise-to-state stability and global asymptotic stability in probability are given, respectively. Finally, the effectiveness of the proposed stability criteria is verified by the simulation results.