Abstract. We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation (1.1) below, for m in the optimal range ((N -2)+/N, 1). Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as t approaches the extinction time. Analogous results are shown for p-laplacian type equations.