Liquid fuel jet in Crossflow (LJIC) is a vital atomization technique significant to the aviation industry. The hydrodynamic instability mechanisms that drive a primary breakup of a transverse jet are investigated using modal and traveling wavelength analysis. This study highlights the primary breakup mechanisms for aviation fuel Jet-A, utilizing a method that could be applied to any liquid fuel. Mathematical decomposition techniques known as POD (Proper Orthogonal Decomposition) and Robust MrDMD (Multi-Resolution Dynamic Mode Decomposition) are used together to identify dominant instability flow dynamics associated with the primary breakup mechanism. Implementation of the Robust MrDMD method deconstructs the nonlinear dynamical systems into multiresolution time-scaled components to capture the intermittent coherent structures. The Robust MrDMD, in conjunction with the POD method, is applied to data points taken across the entire spray breakup regimes: enhanced capillary breakup, bag breakup, multimode breakup, and shear breakup. The dominant frequencies of breakup mechanisms are extracted and identified. These coherent structures are classified with an associated time scale and Strouhal number. Three primary breakup mechanisms, namely ligament shedding, bag breakup, and shear breakup, were identified and associated with the four breakup regimes outlined above. Further investigation portrays these breakup mechanisms to occur in conjunction with each other in each breakup regime, excluding the low Weber number Enhanced Capillary Breakup regime. Spectral analysis of the Robust MrDMD modes' entire temporal window reveals that while multiple breakup mechanisms are convolved, there is a dominant breakup route for each breakup regime. An associated particular traveling wavelength analysis further investigates each breakup mechanism. Lastly, this study explores the effects of an increased momentum flux ratio on each breakup mechanism associated with a breakup regime.