  We show in experiments that a long, underdense, relativistic proton bunch
propagating in plasma undergoes the oblique instability, that we observe as
filamentation. We determine a threshold value for the ratio between the bunch
transverse size and plasma skin depth for the instability to occur. At the
threshold, the outcome of the experiment alternates between filamentation and
self-modulation instability (evidenced by longitudinal modulation into
microbunches). Time-resolved images of the bunch density distribution reveal
that filamentation grows to an observable level late along the bunch,
confirming the spatio-temporal nature of the instability. We calculate the
amplitude of the magnetic field generated in the plasma by the instability and
show that the associated magnetic energy increases with plasma density.
