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The evolution of the secondary vortex arrangement around a foil, performing heaving and pitching motion, is numerically examined for a range of phase offsets (90 ° ≤ ϕ ≤ 270°) and reduced frequency (0.32 ≤ S t c ≤ 0.56), at a Reynolds number of 8000. The wake is dominated by two distinct systems of secondary hairpin-like structures. The first vortex system is associated with an elliptic instability, prompted by the paired primary and secondary leading edge vortices (LEV), which remains persistent within the entire range of Stc. However, the growth of the second system is more closely associated with undulations of the primary LEV at S t c ≥ 0.40, which amplifies as it sheds downstream of the trailing edge. The characteristic presence of the first system is directly linked to the growth of the secondary LEV, formed due to the large-scale interactions under localized adverse pressure gradients. These features promote a streamwise flow compression in neighboring regions of the primary LEV. Subsequently, the stability of these wake arrangements is evaluated using dynamic mode decomposition technique, which identifies a neutrally stable state at the fundamental forcing frequency and its harmonics. However, an increase in the modulus of the first three modes, as Stc increases, coincides with stronger three-dimensionality in the wake. The fundamental mode characterizes the roller undulations, while subsequent modes reveal that the two systems of secondary hairpin-like structures are closely associated with the harmonics of forcing frequency. [ABSTRACT FROM AUTHOR]