  In the delayed explosion scenario of core-collapse supernovae (SNe), the
accretion phase shows pronounced convective overturns and a low-multipole
hydrodynamic instability, the standing accretion shock instability (SASI).
These effects imprint detectable fast time variations on the emerging neutrino
flux. Among existing detectors, IceCube is best suited to this task, providing
an event rate of ~1000 events per ms during the accretion phase for a fiducial
SN distance of 10 kpc, comparable to what could be achieved with a megaton
water Cherenkov detector. If the SASI activity lasts for several hundred ms, a
Fourier component with an amplitude of 1% of the average signal clearly sticks
out from the shot noise. We analyze in detail the output of axially symmetric
hydrodynamical simulations that predict much larger amplitudes up to
frequencies of a few hundred Hz. If these models are roughly representative for
realistic SNe, fast time variations of the neutrino signal are easily
detectable in IceCube or future megaton-class instruments. We also discuss the
information that could be deduced from such a measurement about the physics in
the SN core and the explosion mechanism of the SN.
