Hypercyclicity of C0-semigroups is a very unstable property: We give examples to
show that adding arbitrary small constants or a bounded rank one operator to the generator of a
hypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (even
in operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limit
of nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, the
restriction of a hypercyclic nonnegative semigroup in a Banach lattice to the positive cone may be
far from hypercyclic.