The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable andWiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.