Abstract: Quantum thermal transport in armchair and zigzag graphene nanoribbons is investigated in the presence of single atomic vacancies and subject to different boundary conditions. We start with a full comparison of the phonon polarizations and energy dispersions as given by a fifth-nearest-neighbor force-constant model (5NNFCM) and by elasticity theory of continuum membranes (ETCM). For free-edge ribbons, we discuss the behavior of an additional acoustic edge-localized flexural mode, known as fourth acoustic branch (4ZA), which has a small gap when it is obtained by the 5NNFCM. Then, we show that ribbons with supported edges have a sample-size dependent energy gap in the phonon spectrum which is particularly large for in-plane modes. Irrespective to the calculation method and the boundary condition, the dependence of the energy gap for the low-energy optical phonon modes against the ribbon width W is found to be proportional to 1/W for in-plane, and 1/W-2 for out-of-plane phonon modes. Using the 5NNFCM, the ballistic thermal conductance and its contributions from every single phonon mode are then obtained by the nonequilibrium Green's function technique. We found that, while edge and central localized single atomic vacancies do not affect the low-energy transmission function of in-plane phonon modes, they reduce considerably the contributions of the flexural modes. On the other hand, in-plane modes contributions are strongly dependent on the boundary conditions and at low temperatures can be highly reduced in supported-edge samples. These findings could open a route to engineer graphene based devices where it is possible to discriminate the relative contribution of polarized phonons and to tune the thermal transport on the nanoscale.