Laminar Separation Bubbles (LSBs) may adversely affect aerodynamic performance by causing laminar-turbulent transition. The stability and dynamics of flat plate pressure-gradient-induced LSBs have been studied extensively in the literature; however, surface-induced LSBs, especially those arising from wavy surfaces, have received less attention. Surface waviness, by imposing successive adverse and favourable pressure gradient regions on the wall, creates a complex flow regime, and even can distort the streamlines of flow in a way that affects the stability characteristics of LSBs and triggers three dimensional instabilities different to the ones present in flat plate pressure-gradient-induced LSBs. In this study, the onset of flow separation for numerous sinusoidal surface waviness at various Reynolds numbers is identified. A scaling law is then established, enabling the prediction of the geometric parameters leading to flow separation at a specific Reynolds number. Then, several different cases are selected for further two- and three- dimensional stability analysis using linear and non-linear frameworks. Regarding two-dimensional global linear analysis, an extensive study on effects of domain size and boundary conditions is carried out, and it is shown that choosing the proper domain size and boundary conditions is essential to recover the growth rate of two-dimensional planar waves accurately. To investigate the instabilities of three-dimensional disturbances in a spanwise homogenous baseflow, a two-dimensional global stability approach is utilized, and the resulting large eigenvalue problem is solved using parallel sparse linear algebra techniques. It is shown that, similar to pressure-gradient-induced LSBs on flat plates, three-dimensional instabilities are triggered prior to two dimensional ones. Interestingly, as the curvature of surface waviness increases, new zero-frequency three-dimensional unstable modes localized around the rear part of LSB, not observed in flat plate pressure-gradient-induced LSBs, with relatively large exponential growth rates become the dominant primary instability. Moreover, it is observed that as the curvature increases, the zero-frequency unstable mode that exists for flat-plate pressure-gradient-induced separation bubbles becomes stable and the new localized modes become unstable. Further, three-dimensional DNS will be carried out to identify the transition mechanism for different cases with different curvatures.