This research has received funding from the UK Science and Technology Facilities Council and the European Union Horizon 2020 research and innovation programme (grant agreement No. 647214), and also from JSPS KAKENHI Grant Numbers 25220703 (PI: S. Tsuneta) and 15H03640 (PI: T. Yokoyama). T.V.D. was supported by FWO Vlaanderen’s Odysseus programme, GOA-2015-014 (KU Leuven) and the IAP P7/08 CHARM (Belspo). ; MHD waves permeate the solar atmosphere and constitute potential coronal heating agents. Yet, the waves detected so far may be but a small subset of the true existing wave power. Detection is limited by instrumental constraints, but also by wave processes that localise the wave power in undetectable spatial scales. In this study we conduct 3D MHD simulations and forward modelling of standing transverse MHD waves in coronal loops with uniform and non-uniform temperature variation in the perpendicular cross-section. The observed signatures are largely dominated by the combination of the Kelvin-Helmholtz instability (KHI), resonant absorption and phase mixing. In the presence of a cross-loop temperature gradient we find that emission lines sensitive to the loop core catch different signatures than those more sensitive to the loop boundary and the surrounding corona, leading to an out-of-phase intensity modulation produced by the KHI mixing. Common signatures to all considered models include an intensity and loop width modulation at half the kink period, fine strand-like structure, a characteristic arrow-shaped structure in the Doppler maps, overall line broadening in time but particularly at the loop edges. For our model, most of these features can be captured with a spatial resolution of 0.33″ and spectral resolution of 25 km s-1, although severe over-estimation of the line width is obtained. Resonant absorption leads to a significant decrease of the observed kinetic energy from Doppler motions over time, which is not recovered by a corresponding increase in the line width from phase mixing and the KHI motions. We estimate this hidden wave energy to be a factor of 5-10 of the observed value. ; Postprint ; Peer reviewed