Melting ice made with tap water I recently froze some tap water in plastic bottles and I later made the ice melt completely (the idea was keeping ice bottles nearby me to fight the heat). When the water returned to liquid a lot of tiny white dust/crystal flakes were visible. These were heavier than water and tended to sink.
At the beginning I thought they were little chunks of plastic that got detached from the walls of the bottle due to the severe changes in temperature, but then by comparing the state of the bottles (seemingly unaltered wrt the beginning) with the quantity of flakes I concluded it had to be something else.
The second hypothesis was that the flakes were actually crystals from the minerals that are in the tap water (I hear that in my region tap water is full of calcium for instance). However I do not understand how would these crystals form: 


*

*how could it be possible? and why don't they return/remain diluted in the water?

*do they form during the freezing or the melting phase?


I would say that during the freezing phase the fact that the solution freezes at lower temperatures than pure water could isolate portions of liquid with higher and higher concentration of minerals as the freezing goes on. These will remain mobile till the very end of the freezing process. In the final stage the space for the liquid part to move around will be very limited, giving chance to form crystals. But maybe I am wrong and there is something I am missing.
So any hypothesis on how these crystals form? Maybe they are not even crystals like I am trying to say?
 A: 
So any hypothesis on how these crystals form?

By dropping the temperature, the solubility of some salts in water drops, which could lead to the formation of crystals.
Check out the article Saturated and Supersaturated Solutions.

why don't they return/remain diluted in the water?

Let the water reach room temperature and give it some time. The crystals should dissolve back in the water.
A: If your water source is (for instance) well water, the mineral content will be 
determined by the subsoil composition and temperature near  the well.   Chilling the water can cause
dissolved minerals to precipitate out, as you suggest, BUT the rate of precipitation
will determine the size of the crystals formed (slow chilling should make
few large crystals, fast chilling should make many small crystals)
The rate at which warmed water will dissolve those crystals is proportional to
the exposed area (on the order of the square of the crystal diameter), but the
mass of the crystals is proportional to the volume (cube of the crystal
diameter).   So, the persistance of precipitated material after warming could
depend on 


*

*the 'warm' temperature relative to the subsoil temperature (presumed to be equilibrium for the mineral content)

*the rate at which the tap water was chilled

*the time after warming that the precipitate was observed

*any chemical changes (outgassing of dissolved CO2, for instance, can de-acidify the water)

