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Given a Polish topology $\tau$ on ${{\mathcal{B}}_{1}(X)}$, the set of allcontraction operators on $X=\ell_p$, $1\le p<\infty$ or $X=c_0$, we proveseveral results related to the following question: does a typical $T\in{{\mathcal{B}}_{1}(X)}$ in the Baire Category sense has a non-trivial invariantsubspace? 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 anon-trivial invariant subspace? We mostly focus on the Strong Operator Topologyand the Strong$^*$ Operator Topology.

Does a typical $\ell_p\,$-$\,$space contraction have a non-trivial invariant subspace?