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

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.