  The property of cyclicity of a linear operator, or equivalently the property
of simplicity of its spectrum, is an important spectral characteristic that
appears in many problems of functional analysis and applications to
mathematical physics. In this paper we study cyclicity in the context of
rank-one perturbation problems for self-adjoint and unitary operators. We show
that for a fixed non-zero vector the property of being a cyclic vector is not
rare, in the sense that for any family of rank-one perturbations of
self-adjoint or unitary operators acting on the space, that vector will be
cyclic for every operator from the family, with a possible exception of a small
set with respect to the parameter. We discuss applications of our results to
Anderson-type Hamiltonians.
