<p>Compression corner flows exhibit a shock-wave/boundary-layer interaction that induce flow separation and potential laminar-to-turbulent transition that may produce high surface heating in the reattachment region. Laminar separated flows can sustain the amplification of global instabilities that can be the origin for boundary-layer transition. A computational study of double wedge and cone-flare geometries at zero degrees angle of attack is performed to investigate the effects of geometrical parameters, wall temperature, and freestream conditions on the laminar separation and its global instability characteristics, with the objective of deriving a simple correlation between the separated flow properties and the global instability onset. Regardless of the nosetip/leading edge bluntness, the global instability is first dominated by a stationary mode with low azimuthal wavenumber that extends across the entire separated region. To examine the correlation between separation bubble strength and the onset of global instability, we created and analyzed a database spanning a wide range of parameters, including freestream Mach number, Reynolds number, surface temperature, nosetip/leading edge radius, an intermediate cylindrical section ahead of the flare, radius of curvature at the cone-flare corner, and flow turning angles. Multiple normalized quantities are investigated as potential metrics toward an empirical criterion for global instability onset. Generally, the separation bubble becomes globally unstable when the maximum reverse streamwise velocity exceeds 10% of the freestream speed. This work provides a computational overview of how each flow and geometrical parameter affects the laminar separation regions, guiding future experimental studies to distinguish whether observed flow trends result from global instability and potential laminar-to-turbulent transition prior to reattachment.</p>