  We systematically explore the spectrum of gravitational perturbations in
codimension-1 DGP braneworlds, and find a 4D ghost on the self-accelerating
branch of solutions. The ghost appears for any value of the brane tension,
although depending on the sign of the tension it is either the helicity-0
component of the lightest localized massive tensor of mass $0<m^2 < 2H^2$ for
positive tension, the scalar `radion' for negative tension, or their admixture
for vanishing tension. Because the ghost is gravitationally coupled to the
brane-localized matter, the self-accelerating solutions are not a reliable
benchmark for cosmic acceleration driven by gravity modified in the IR. In
contrast, the normal branch of solutions is ghost-free, and so these solutions
are perturbatively safe at large distance scales. We further find that when the
$\mathbb{Z}_2$ orbifold symmetry is broken, new tachyonic instabilities, which
are much milder than the ghosts, appear on the self-accelerating branch.
Finally, using exact gravitational shock waves we analyze what happens if we
relax boundary conditions at infinity. We find that non-normalizable bulk
modes, if interpreted as 4D phenomena, may open the door to new ghost-like
excitations.
