  We study energy decay rates for the damped wave equation with unbounded
damping, without the geometric control condition. Our main decay result is
sharp polynomial energy decay for polynomially controlled singular damping on
the torus. We also prove that for normally $L^p$-damping on compact manifolds,
the Schr\"odinger observability gives $p$-dependent polynomial decay, and
finite time extinction cannot occur. We show that polynomially controlled
singular damping on the circle gives exponential decay.
