Finding a robust and efficient solver for (non-)symmetric systems that arise in incompressible Computational Fluid Dynamics (CFD) is of great interest to both academia and industry. We consider the Compatible Discrete Operator (CDO) discretization that has recently been devised for CFD simulations in the context of incompressible Stokes and Navier-Stokes flows. The discrete problems resulting from CDO schemes yield large saddle-point systems that require relevant numerical methods suitable to deal with large indefinite and poorly conditioned linear systems. In this paper, we focus on two segregated methods: the augmented Lagrangian Uzawa method and the generalized Golub-Kahan bidiagonalization, as well as a monolithic method based on an algebraic transformation by change of variables. We also employ algebraic multigrid (AMG) preconditioned Krylov solvers such as the Flexible Conjugate Gradient (FCG) method, and the Flexible Generalized Minimal Residual (FGMRES) method, to solve the linear systems. Using the CFD software code saturne, we compare the numerical performance with respect to the choice of linear solvers and numerical strategies for the saddle-point problem. In the numerical experiments, the AMG preconditioned Krylov methods show robustness in test cases of Stokes and Navier-Stokes problems.