If I have a linear system, I can check with the following procedure. Consider the linear system:
$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$$ $$\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u} $$
I can check if the unobservable and uncontrollable modes are detectable and stabilizable by looking at the eigenvalue decomposition i.e., $\mathbf{A} = \mathbf{W} \mathbf{\Lambda} \mathbf{V}$ of the system.
I can check if $\mathbf{C}\mathbf{W}$ has any zero columns and if $\mathbf{V} \mathbf{B}$ has any zero rows. If so, the zero columns and zero rows correspond to the unobservable and uncontrollable modes. However, even if unobservable/uncontrollable, I can check if the eigenvalues are in the LHP (left hand plane), and if so, the modes are detectable/stabilizable respectively.
How can I check for nonlinear systems? I can linearize, but that requires some linearization point, and in that case, I will only be able to check if stabilizable/detectable about some point. Also, I don't know if that is even a valid approach for some specific point.