Summary The standard Fluid‐Structure Interaction ( fsi ) coupling, that uses as unknowns velocity and pressure for the fluid and displacements for the solid, is compared against two novel types of coupling, the first one a three‐field coupling (velocity‐pressure‐stress/displacement‐pressure‐stress) introduced by the authors in a recent work, and a two‐field coupling (velocity‐pressure/displacement‐pressure) introduced in this paper, in this way completing our set of Field to Field ( f2f ) equations, all stabilized by means of the Variational Multi‐Scale ( vms ) method using dynamic and orthogonal subscales. The solid two‐field fsi coupling formulation is benchmarked statically and dynamically. Proper Orthogonal Decomposition ( pod ) is applied to all three fsi formulations to obtain reduced basis and asses their performance in a reduced space. Numerical tests are shown comparing all three formulations. By correctly resolving the Cauchy stress tensor, the three‐field fsi coupling proves to provide more accurate results in both Full Order Model ( fom ) and Reduced Order Model ( rom ) spaces than its counterparts for a similar number of degrees of freedom, making it a reliable formulation. f2f pairing appears to be beneficial, providing more accurate results in all cases shown; mixed pairing with a three‐field formulation in the solid appears to produce very precise results as well.