  Using Read's construction of operators without non-trivial invariant
subspaces/subsets on $\ell_{1}$ or $c_{0}$, we construct examples of operators
on a Hilbert space whose set of hypercyclic vectors is "large" in various
senses. We give an example of an operator such that the closure of every orbit
is a closed subspace, and then, answering a question of D. Preiss, an example
of an operator such that the set of its non-hypercyclic vectors is Gauss null.
This operator has the property that it is orbit-unicellular, i.e. the family of
the closures of its orbits is totally ordered. We also exhibit an example of an
operator on a Hilbert space which is not orbit-reflexive.
