We analyse the three–dimensional non–parallel instability mechanisms responsible for transition to turbulence in regions of recirculating steady laminar two–dimensional incompressible separation bubble flow in a twofold manner. First, we revisit the problem of Tollmien–Schlichting (TS)–like disturbances and we demonstrate, for the first time for this type of flow, excellent agreement between the parabolized stability equation results and those of independently performed direct numerical simulations. Second, we perform a partial–derivative eigenvalue problem stability analysis by discretizing the two spatial directions on which the basic flow depends, precluding TS–like waves from entering the calculation domain. A new two–dimensional set of global amplified instability modes is thus discovered. In order to prove earlier topological conjectures about the flow structural changes occurring prior to the onset of bubble unsteadiness, we reconstruct the total flowfield by linear superposition of the steady two–dimensional basic flow and the new most–amplified global eigenmodes. In the parameter range investigated, the result is a bifurcation into a three–dimensional flowfield in which the separation line remains unaffected while the primary reattachment line becomes three dimensional, in line with the analogous result of a multitude of experimental observations.