Propagating atomic vibrational waves—phonons—determine important thermal, mechanical, optoelectronic and transport characteristics of materials. Thus a knowledge of phonon dispersion (that is, the dependence of vibrational energy on momentum) is a key part of our understanding and optimization of a material's behaviour. However, the phonon dispersion of a free-standing monolayer of a two-dimensional material such as graphene, and its local variations, have remained elusive for the past decade because of the experimental limitations of vibrational spectroscopy. Even though electron energy loss spectroscopy (EELS) in transmission has recently been shown to probe local vibrational charge responses1-4, such studies are still limited by momentum space integration due to the focused beam geometry; they are also restricted to polar materials such as boron nitride or oxides1-4, in which huge signals induced by strong dipole moments are present. On the other hand, measurements on graphene performed by inelastic X-ray (neutron) scattering spectroscopy5-7 or EELS in reflection8,9 do not have any spatial resolution and require large microcrystals. Here we provide a new pathway to determine phonon dispersions down to the scale of an individual free-standing graphene monolayer by mapping the distinct vibrational modes for a large momentum transfer. The measured scattering intensities are accurately reproduced and interpreted with density functional perturbation theory10. Additionally, a nanometre-scale mapping of selected momentum-resolved vibrational modes using graphene nanoribbon structures has enabled us to spatially disentangle bulk, edge and surface vibrations. Our results are a proof-of-principle demonstration of the feasibility of studying local vibrational modes in two-dimensional monolayer materials at the nanometre scale.