  The simplest flux compactifications are highly symmetric---a $q$-form flux is
wrapped uniformly around an extra-dimensional $q$-sphere. In this paper, we
investigate solutions that break the internal SO($q+1$) symmetry down to
SO$(q)\times\mathbb Z_2$; we find a large number of such lumpy solutions, and
show that often at least one of them has lower vacuum energy, larger entropy,
and is more stable than the symmetric solution. We construct the phase diagram
of lumpy solutions, and provide an interpretation in terms of an effective
potential. Finally, we provide evidence that the perturbatively stable vacua
have a non-perturbative instability to spontaneously sprout lumps; we give an
estimate of the decay rate and argue that generically it is exponentially
faster than all other known decays.
