  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 $0.33\arcsec$ 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.
