  We investigate the many-body instabilities of electrons interacting near Van
Hove singularities arising in monolayer and twisted bilayer graphene. We show
that a pairing instability must be dominant over the tendency to magnetic order
as the Fermi level is tuned to the Van Hove singularity in the conduction band
of graphene. As a result of the extended character of the saddle points in the
dispersion, we find that the pairing of the electrons takes place
preferentially in a channel of f-wave symmetry, with an order parameter
vanishing at the position of the saddle points along the Fermi line. In the
case of the twisted bilayers, the dispersion has instead its symmetry reduced
down to the C_{3v} group and, most importantly, it leads to susceptibilities
that diverge at the saddle points but are integrable along the Fermi line. This
implies that a ferromagnetic instability becomes dominant in the twisted
graphene bilayers near the Van Hove singularity, with a strength which is
amplified as the lowest subband of the electron system becomes flatter for
decreasing twist angle.
