  We study the critical behaviour of Anderson localized modes near intersecting
flat and dispersive bands in the quasi-one-dimensional diamond ladder with weak
diagonal disorder $W$. The localization length $\xi$ of the flat band states
scales with disorder as $\xi \sim W^{-\gamma}$, with $\gamma \approx 1.3$, in
contrast to the dispersive bands with $\gamma =2$. A small fraction of
dispersive modes mixed with the flat band states is responsible for the unusual
scaling. Anderson localization is therefore controlled by two different length
scales. Nonlinearity can produce qualitatively different wave spreading
regimes, from enhanced expansion to resonant tunneling and self-trapping.
