This paper is concerned with the analysis and verification of stability for a class of stochastic hybrid systems (SHSs), which contain not only continuous flows described by stochastic differential equations but also Markovian switching and discrete jumps together. To start with, taking advantage of multiple Lyapunov functions (MLFs), we propose a series of sufficient conditions for (uniform) stability in probability, (uniform) asymptotic stability in probability, stochastically asymptotic stability in the large and (uniform) exponential stability in probability, respectively, in which indefinite scalar functions are utilized to relax the conservativeness of the classical MLFs. Then, based on the generalized exponential martingale inequality for Levy type stochastic integral, we extend our less conservative MLFs and indefinite scalar functions based method to obtain a sufficient condition for globally almost sure exponential stability. Further, we investigate the mechanical verification of all the above stabilities for rational SHSs by proposing a semidefinite programming based computational approach. Especially, our mechanical verification approach can also be applied to deal with certain systems beyond rational SHSs. The applicabilities of both our theoretical results and mechanical approach are demonstrated by academic examples.