This paper presents significant improvements to the nominal control scheme of the open-access Solo-12 quadruped and an experimental comparative study of different Model Predictive Controllers (MPC) that were implemented and tested on the robot. The modifications of the controller formulation that improved the nominal behavior and reduced tracking oscillations are first described. Thanks to them, the maximum reachable velocity of the robot was doubled and the foot placement was improved. On this basis three MPCs of increasing complexity were compared to evaluate the validity of different assumptions and heuristics. They range from a simplified linear centroidal model with contact points fixed by a heuristics, to a nonlinear one that also optimizes the contact points locations. Experimental results show that, thanks to the fast solver Crocoddyl and the proposed formulation, similar performances can be obtained with a MPC that optimizes both the center of mass trajectory and the foot placement, while taking into accounts the nonlinearity of the centroidal model, than with a modular scheme using a heuristic for foot placement and considering a linearized model. This paves the way for future work such as leveraging information about the environment to improve footsteps placement and timings.