  We report on quantum capacitance measurements of high quality, graphite- and
hexagonal boron nitride encapsulated Bernal stacked trilayer graphene devices.
At zero applied magnetic field, we observe a number of electron density- and
electrical displacement-tuned features in the electronic compressibility
associated with changes in Fermi surface topology. At high displacement field
and low density, strong trigonal warping gives rise to emergent Dirac gullies
centered near the corners of the hexagonal Brillouin and related by three fold
rotation symmetry. At low magnetic fields of $B=1.25$~T, the gullies manifest
as a change in the degeneracy of the Landau levels from two to three. Weak
incompressible states are also observed at integer filling within these
triplets Landau levels, which a Hartree-Fock analysis indicates are associated
with Coulomb-driven nematic phases that spontaneously break rotation symmetry.
