  Simon's subshift conjecture states that for every aperiodic minimal subshift
of Verblunsky coefficients, the common essential support of the associated
measures has zero Lebesgue measure. We disprove this conjecture in this paper,
both in the form stated and in the analogous formulation of it for discrete
Schr\"odinger operators. In addition we prove a weak version of the conjecture
in the Schr\"odinger setting. Namely, under some additional assumptions on the
subshift, we show that the density of states measure, a natural measure
associated with the operator family and whose topological support is equal to
the spectrum, is singular. We also consider one-frequency quasi-periodic
Schr\"odinger operators with continuous sampling functions and show that
generically, the density of states measure is singular as well.
