For a power bounded or polynomially bounded operator T sufficient conditions for the
existence of a nontrivial hyperinvariant subspace are given. These hyperinvariant
subspaces are the closures of the range of phi(T), where phi is a singular inner function
if T is polynomially bounded, or phi is a function analytic in the unit disc with
absolutely summable Taylor coefficients and singular inner part if T is supposed to
be only power bounded. Also, an example of a quasianalytic contraction T is given
such that the quasianalytic spectral set of T is not the whole unit circle T, while
sigma(T) = T. The proofs are based on results by Esterle, Kellay, Borichev and Volberg.