It has been observed that small stochastic perturbations in dynamical systems exhibiting stable limit cycle attractors lead to the trajectories forming a stochastic bundle around the trajectory of the corresponding noise free system.For systems with multiple attractors, these stochastic perturbations may result in intermittent transitions between co-existing stable states.This talk will focus on quantifying the likelihood of these transitions in terms of the local stability of the stochastic limit cycle attractor. The developments are illustrated through a Rossler system, a parametrically forced Duffing oscillator and a non-smooth system comprising of a soft-impacting oscillator. The effect of shift in phase dynamics due to the stochastic perturbations will be addressed. Measures of local and global stability of stochastic dynamical systems using sample-based approaches like basin stability and survivability will be discussed. A method for numerical characterization of the basins of attraction for noisy dynamical systems will be proposed and the notion of basin erosion as a function of noise intensity will be discussed.