Given a Polish topology
τ
\tau
on
B
1
(
X
)
\mathcal {B}_{1}(X)
, the set of all contraction operators on
X
=
ℓ
p
X=\ell _p
,
1
≤
p
>
∞
1\le p>\infty
or
X
=
c
0
X=c_0
, we prove several results related to the following question: does a typical
T
∈
B
1
(
X
)
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
δ
G_\delta
set
G
⊆
(
B
1
(
X
)
,
τ
)
\mathcal G\subseteq (\mathcal {B}_{1}(X),\tau )
such that every
T
∈
G
T\in \mathcal G
has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong
∗
^*
Operator Topology.