  Supernova neutrinos have several exceptional features which can lead to
interesting physical consequences. At the production point their wave packets
have an extremely small size $\sigma_x \sim 10^{-11}$ cm; hence the energy
uncertainty can be as large as the energy itself, $\sigma_E \sim E$, and the
coherence length is short. On the way to the Earth the wave packets of mass
eigenstates spread to macroscopic sizes and separate. Inside the Earth the mass
eigenstates split into eigenstates in matter and oscillate again. The coherence
length in the Earth is comparable with the radius of the Earth. We explore
these features and their consequences. (i) We present new estimates of the wave
packet size. (ii) We consider the decoherence condition for the case of wave
packets with spatial spread and show that it is not modified by the spread.
(iii) We study the coherence of neutrinos propagating in a multi-layer medium
with density jumps at the borders of layers. In this case coherence can be
partially restored due to a "catch-up effect", increasing the coherence length
beyond the usual estimate. This catch-up effect can occur for supernova
neutrinos as they cross the shock wave fronts in the exploding star or the core
of the Earth.
