  Given a Polish topology $\tau$ on ${{\mathcal{B}}_{1}(X)}$, the set of all
contraction operators on $X=\ell_p$, $1\le p<\infty$ or $X=c_0$, we prove
several results related to the following question: does a typical $T\in
{{\mathcal{B}}_{1}(X)}$ in the Baire Category sense has a non-trivial invariant
subspace? In other words, is there a dense $G_\delta$ set $\mathcal G\subseteq
({{\mathcal{B}}_{1}(X)},\tau)$ such that every $T\in\mathcal G$ has a
non-trivial invariant subspace? We mostly focus on the Strong Operator Topology
and the Strong$^*$ Operator Topology.
