The identification of nonlinear systems in aeroelasticity poses a significant challenge for practitioners, often hampered by the complex nature of aeroelastic response data which may contain multiple forms of nonlinearity. Characterizing and quantifying nonlinearity is further complicated when the dynamic oscillations are of a high–amplitude limit cycle form, masking the underlying nonlinear contributions. In the present paper, a three– degree–of–freedom airfoil with freeplay in the control surface and transonic aerodynamics is investigated. The main form of analysis is via higher–order spectra to unveil the form of nonlinearity that i) freeplay with linearized aerodynamics and ii) freeplay with nonlinear inviscid aerodynamic phenomena will produce. It is shown that the freeplay nonlinearity with linearized aerodynamics is characterized solely by cubic interactions, i.e., the quadratic interactions are negligible. However, when considering the Euler-based CFD simulations, as the amplitude of the of the oscillations increases and Type-B shock motion becomes more apparent, the strength of the quadratic interactions becomes prominent. The findings demonstrate how the interaction between the different types of nonlinearity and the different solution methods affect the nonlinear spectral content of the system and how different forms of nonlinearity can be characterized by their higher–order spectra.