Does every compact complex manifold of complex dimension greater than or equal two possess a nontrivial rank 2 holomorphic vector bundle?
Some partial answer:
On non-Kahler surfaces we can give stronger version of your question and we have allways the following thorem
Theorem: Let $S$ be non-Kahler surfaces, For any $n > 0$ there exists a Mumford stable rank-2 vector bundle $E$ with trivial determinant, ($det(E) = \mathcal O_S$ ) and $c_2(E)=n$.
We have also the following theorem due to Banica - Le Potier
Theorem: If $S$ is a non-projective surface and $N$ is enough big , then there exists rank-2 vector bundles on $S$ with $c_2(E) = N$ which are non-filtrable.
We have following theorem due to Ballico and Schwarzenberger
Theorem: If $S$ be a projective surface and $E$ is any topological rank-2 vector bundle with $det(E) \in Pic(X)$, then $E$ has a holomorphic structure by applying elementary transformations starting from the trivial bundle.
We have Cartan-Serre theorem for construction of rank 2 vector bundles on projetive varieties.
Theorem: Let $X$ be a complex manifold, $L_1, L_2 ∈ Pic(X)$ are line bundles, $Z ⊂ X$ with $codim_X(Z) = 2$. Under some cohomological conditions, the sheaf $E$ sitting in
$$0 \to L_1 \to E \to L_2 \otimes J_Z → o$$ is locally free. Moreover, If $X$ is projective, any rank-2 holomorphic vector bundle can be constructed this way