  We study discrete surface solitons in semi-infinite, one-dimensional,
nonlinear (Kerr), quasiperiodic waveguide arrays of the Fibonacci and
Aubry-Andr\'e types, and explore different families of localized surface modes,
as a function of optical power content (`nonlinearity') and quasiperiodic
strength (`disorder'). We find a strong asymmetry in the power content of the
mode as a function of the propagation constant, between the cases of focussing
and defocussing nonlinearity, in both models. We also examine the dynamical
evolution of a completely-localized initial excitation at the array surface. We
find that in general, for a given optical power, a smaller quasiperiodic
strength is required to effect localization at the surface than in the bulk.
Also, for fixed quasiperiodic strength, a smaller optical power is needed to
localize the excitation at the edge than inside the bulk.
