  We consider discrete one-dimensional Schr\"odinger operators with
quasi-Sturmian potentials. We present a new approach to the trace map dynamical
system which is independent of the initial conditions and establish a
characterization of the spectrum in terms of bounded trace map orbits. Using
this, it is shown that the operators have purely singular continuous spectrum
and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a
subclass having purely $\alpha$-continuous spectrum. All these results hold
uniformly on the hull generated by a given potential.
