The optical local-field enhancement on nanometer length scales provides the basis for plasmonic metal nanostructures to serve as molecular sensors and as nanophotonic devices. However, particle morphology and the associated surface plasmon resonance alone do not uniquely reflect the important details of the local field distribution. Here, we use interferometric homodyne tip-scattering near-field microscopy for plasmonic near-field imaging of crystalline triangular silver nanoprisms. Strong spatial field variation on lengths scales as short as 20 nm are observed sensitively depending on structural details and environment. The poles of the dipole and quadrupole plasmon modes, as identified by phase-sensitive probing and calculations performed in the discrete dipole approximation (DDA), reflect the particle symmetry. Together with the observation that the largest enhancement is not necessarily found to be associated with the tips of the nanoprisms, our results provide critical information for the selection of particle geometries as building blocks for plasmonic device applications.