When can water be supercooled? Qualitatively, I understand that water can be supercooled when:


*

*It is relatively pure.

*It is in a container that is relatively smooth of defects.


The effect of both of these is to reduce nucleation points, which are needed to provide a place for the ice crystals to start growing.
Rate of cooling may also be a factor- this is less clear to me.
But.... surely water that is experimentally supercooled is not perfectly free of impurities, nor is its container smooth at the atomic level. So there must be some critical amount of nucleation sites available.
Is it possible to quantify somehow, whether in general or at least for a specific impurity, exactly what the critical amount is to prevent supercooling?  Is it a critical density of these sites that matters, or just a critical total number, since each site has some probability of starting crystallization? Even better, is there some general energetic or thermodynamic inequality that describes what conditions are needed for successful supercooling?
 A: 
Qualitatively, I understand that water can be supercooled when:



*

*It is relatively pure.

*It is in a container that is relatively smooth of defects.

*...


Yes, but it's actually very complicated ...


See here for a huge chart of ice types - this site seems to be down, with the last complete capture by Wayback on Oct. 9 2020); some later captures are incomplete, missing some of the images and links to other webpages.

 Phase Diagram




Is it possible to quantify somehow, whether in general or at least for a specific impurity, exactly what the critical amount is to prevent supercooling?
...
Even better, is there some general energetic or thermodynamic inequality that describes what conditions are needed for successful supercooling?

It is explained relatively simply at Wikipedia's Supercooling webpage; and at great length, but still relatively simply, at the "Amorphous Ice and Glassy Water" and "Explanation of the Phase Anomalies of Water (P1-P13)" webpages (so, 14 webpages, more than you likely wanted to know).
Conditions such as: cooling rate, impurities, pressure, container, shock waves, all have an effect on the results you obtain, sometimes a bit of luck is involved (something you don't account for, experimental error).
You can make syrup if you do it correctly.

Wikipedia gives a simplified explanation of heterogeneous nucleation which in water usually occurs when a crystal of ice water is added to supercooled water.
Some experimental results were published in the Journal of Physics article "High-density amorphous ice: nucleation of nanosized low-density amorphous ice", the Journal of the American Chemical Society article "Heterogeneous Nucleation of Ice on Carbon Surfaces", and in the Proceedings of the National Academy of Sciences article "Observing the formation of ice and organic crystals in active sites".
A: Hopefully I can dig into these a bit more and flesh out the answer, but for now here are some relevant resources & quotes:
"Effect of solutes on the heterogeneous nucleation temperature
of supercooled water: an experimental determination" Physical Chemistry Chemical Physics (2009):

Homogeneous nucleation 
  Homogeneous nucleation occurs only in water not influenced
  by surfaces and devoid of foreign particles or substances. Only
  the water molecules are involved in the freezing event and at
  some homogeneous nucleation temperature, $T_{hom}$, estimated
  to be $\approx -41^\circ$$C^5$ they form an ice-like nucleus, or cluster, large
  enough to then cause spontaneous freezing. The practicalities
  of the experimental determination of this temperature are
  difficult and usually involve an emulsion technique and an
  averaging of the measured $T_{hom}$ values in an attempt to
  smooth out the inherent stochastic nature of nucleation.$^6$
   
Heterogeneous nucleation 
  Ice nucleation can also occur at the surface of so called ‘‘ice
  nuclei’’ by heterogeneous nucleation.7 The nuclei can be dirt,
  large molecules, bacteria, or simply the container wall. In each
  case a specific nucleating surface allows scaling of the free
  energy barrier (in classical theory) and causes the freezing
  event to proceed. The study of heterogeneous nucleation is
  of much more practical importance than homogeneous
  nucleation because most nucleation events in nature are
  heterogeneous.
Effects of solutes on $T_{hom}$ and $T_{het}$
It is well established that for homogeneous nucleation in
  aqueous solutions the lowering of $T_{hom}$ is linearly related to
  solute concentration and is independent of the solute, at least
  for small molecules, i.e.
$\Delta T_{hom} = \lambda \Delta T_m$
$^{8,9}$ The multiplying
  factor $\lambda$ is generally cited as 2.0$^{10,11}$ although there is some
  debate, with values as low as 1.7 being quoted.$^{12}$ We are not
  aware of a molecular explanation of this factor. It has also
  been reported that high molecular weight solutes have a larger
  effect than smaller molecules and that l can reach values as
  high as five for large molecules.$^{13,14}$
  It is clear that the effects of solute on the heterogeneous
  nucleation temperature, $T_{het}$, have wide-ranging consequences
  in areas as diverse as cloud formation,15 ice-cream and other
  foods16 and in freeze-tolerant organisms.17 As an example, if
  sugar is added to ice-cream prior to manufacture the freezing
  point is reduced by B1.86 1C per mole but the nucleation
  temperature is reduced by B3.7 1C, allowing for deeper
  supercooling and so causing more, but smaller, nuclei at the
  time of nucleation, and so smoother ice-cream. Accurate
  determination of $\lambda$ is of importance to all of these fields
  of study.

"Measurements of the concentration and composition
of nuclei for cirrus formation" Proceedings of the National Academy of Sciences (2003):

measurements of the concentration and
  composition of tropospheric aerosol particles capable of initiating
  ice in cold (cirrus) clouds are reported

A: When pure water is cooled below freezing point,it may remain in a super cooled state.Super cooling is a state where liquids do not solidify even below their normal freezing point.
This concept can applied in why water in cloud do not freeze.
