This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (explosion suppression) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations. This work reveals the impact of various random effects on the underlying systems for almost sure and $p$th-moment stability and provides insight on stability and stabilization of switching jump diffusion systems.