  We prove that if T is an operator on an infinite-dimensional Hilbert space
whose spectrum and essential spectrum are both connected and whose Fredholm
index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces
of T are the finite-dimensional ones. We also characterize norm-stable
invariant subspaces of any weighted unilateral shift operator. We show that
quasianalytic shift operators are points of norm continuity of the lattice of
the invariant subspaces. We also provide a necessary condition for strongly
stable invariant subspaces for certain operators.
