  Ferromagnetism is most common in transition metal compounds but may also
arise in low-density two-dimensional electron systems, with signatures observed
in silicon, III-V semiconductor systems, and graphene moir\'e heterostructures.
Here we show that gate-tuned van Hove singularities in rhombohedral trilayer
graphene drive the spontaneous ferromagnetic polarization of the electron
system into one or more spin- and valley flavors. Using capacitance
measurements on graphite-gated van der Waals heterostructures, we find a
cascade of density- and electronic displacement field tuned phase transitions
marked by negative electronic compressibility. The transitions define the
boundaries between phases where quantum oscillations have either four-fold,
two-fold, or one-fold degeneracy, associated with a spin and valley degenerate
normal metal, spin-polarized `half-metal', and spin and valley polarized
`quarter metal', respectively. For electron doping, the salient features are
well captured by a phenomenological Stoner model with a valley-anisotropic
Hund's coupling, likely arising from interactions at the lattice scale. For
hole filling, we observe a richer phase diagram featuring a delicate interplay
of broken symmetries and transitions in the Fermi surface topology. Finally, by
rotational alignment of a hexagonal boron nitride substrate to induce a moir\'e
superlattice, we find that the superlattice perturbs the preexisting isospin
order only weakly, leaving the basic phase diagram intact while catalyzing the
formation of topologically nontrivial gapped states whenever itinerant half- or
quarter metal states occur at half- or quarter superlattice band filling. Our
results show that rhombohedral trilayer graphene is an ideal platform for
well-controlled tests of many-body theory and reveal magnetism in moir\'e
materials to be fundamentally itinerant in nature.
