  Recent developments in the observation and modelling of kink oscillations of
coronal loops have led to heightened interest over the last few years. The
modification of the Transverse Density Profile (TDP) of oscillating coronal
loops by non-linear effects, in particular the Kelvin-Helmholtz Instability
(KHI), is investigated. How this evolution may be detected is established, in
particular, when the KHI vortices may not be observed directly. A model for the
loop's TDP is used which includes a finite inhomogeneous layer and homogeneous
core, with a linear transition between them. The evolution of the loop's
transverse intensity profile from numerical simulations of kink oscillations is
analysed. Bayesian inference and forward modelling techniques are applied to
infer the evolution of the TDP from the intensity profiles, in a manner which
may be applied to observations. The strongest observational evidence for the
development of the KHI is found to be a widening of the loop's inhomogeneous
layer, which may be inferred for sufficiently well resolved loops, i.e $>$ 15
data points across the loop. The main signatures when observing the core of the
loop (for this specific loop model) during the oscillation are: a widening
inhomogeneous layer, decreasing intensity, an unchanged radius, and visible
fine transverse structuring when the resolution is sufficient. The appearance
of these signatures are delayed for loops with wider inhomogeneous layers, and
quicker for loops oscillating at higher amplitudes. These cases should also
result in stronger observational signatures, with visible transverse
structuring appearing for wide loops observed at SDO/AIA resolution.
