The structure of nuclei around the neutron-rich nucleus 132Sn is of particular interest due to the vicinity of the Z=50 and N=82 shell closures and the r-process nucleosynthetic path. Four states in 131Sn with a strong single-particle-like component have previously been studied via the (d,p) reaction, with limited excitation energy resolution. The 130Sn(9Be,8Be)131Sn and 130Sn(13C,12C)131Sn single-neutron transfer reactions were performed in inverse kinematics at the Holifield Radioactive Ion Beam Facility using particle-γ coincidence spectroscopy. The uncertainties in the energies of the single-particle-like states have been reduced by more than an order of magnitude using the energies of γ rays. The previous tentative Jπ values have been confirmed. Decays from high-spin states in 131Sn have been observed following transfer on the isomeric component of the 130Sn beam. The improved energies and confirmed spin-parities of the p-wave states important to the r-process lead to direct-semidirect cross sections for neutron capture on the ground state of 130Sn at 30 keV that are in agreement with previous analyses. A similar assessment of the impact of neutron-transfer on the isomer would require significant nuclear structure and reaction theory input. There are few measurements of transfer reaction on isomers, and this is the first on an isomer in the 132Sn region.