IceCube was completed in December 2010. It forms a lattice of 5160 photomultiplier tubes that monitor a volume of ~ 1 cubic km in the deep Antarctic ice for particle induced photons. The telescope was designed to detect neutrinos with energies greater than 100 GeV. Owing to subfreezing ice temperatures, the photomultiplier dark noise rates are particularly low. Hence IceCube can also detect large numbers of MeV neutrinos by observing a collective rise in all photomultiplier rates on top of the dark noise. With 2 ms timing resolution, IceCube can track subtle features in the temporal development of the supernova neutrino burst. For a supernova at the galactic center, its sensitivity matches that of a background-free megaton-scale supernova search experiment. The sensitivity decreases to 20 standard deviations at the galactic edge (30 kpc) and 6 standard deviations at the Large Magellanic Cloud (50 kpc). IceCube is sending triggers from potential supernovae to the Supernova Early Warning System. The sensitivity to neutrino properties such as the neutrino hierarchy is discussed and simulations of tantalizing signatures, such as the formation of a quark star or a black hole as well as the characteristics of shock waves are presented. All results are preliminary.