We consider various method of fundamental solution (MFS) formulations for the numerical solution of two-dimensional boundary value problems (BVPs) governed by the homogeneous biharmonic equation. The motivation for employing the proposed techniques comes from the corresponding boundary integral representations. These are carefully analyzed in the case the domain of the BVP under consideration is a disk. The results of this analysis detect a potentially troublesome case in one of the proposed MFS approaches. Numerical results confirm the analytical findings for more general domains.