  We consider the supercooled Stefan problem, which captures the freezing of a
supercooled liquid, in one space dimension. A probabilistic reformulation of
the problem allows to define global solutions, even in the presence of blow-ups
of the freezing rate. We provide a complete description of such solutions, by
relating the temperature distribution in the liquid to the regularity of the
ice growth process. The latter is shown to transition between (i) continuous
differentiability, (ii) Holder continuity, and (iii) discontinuity. In
particular, in the second regime we rediscover the square root behavior of the
growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for
the ordinary Stefan problem. In our second main theorem, we establish the
uniqueness of the global solutions, a first result of this kind in the context
of growth processes with singular self-excitation when blow-ups are present.
