I too have little white things floating in my water, but I only see them when I use the tap water to make ice cubes. When the ice cubes (made from filtered water - a Pur water filter attached to the kitchen faucet) melt in a drink, little white pieces of what looks like particles of puréed toilet paper or mucus swirl in the glass and eventually settle on the bottom. This only happens when the water is frozen and then melted. There is no debris visible in the water coming out of the Pur Filter, or even the straight out of the faucet.

I have no clue where to begin looking for identification or solution to the problem. It really is gross to drink half a glass of water and then see things swirling around in it.

Did you ever find an explanation to this problem (or a solution)? We just moved into a house and have the same problem with the ice cubes. It is disgusting! The water company says it is calcium deposits that come together when the water is frozen, but this is not what I would imagine calcium deposits to look like. Your description is exactly right... it looks like tiny little pieces of toilet paper. I'd love to hear if you've found any answers!

I am checking on this with a scientific research group. I have the same effect in my tap water and ice cube process as well. I will post their answer as soon as I hear back from them. As of right now my guess is that the particles are calcium hydroxide but will know for sure shortly.

So the research group consensus is this:
Tap water has about 100-200 ppmw (parts-per-million, by weight) of dissolved minerals. Crystallization of water as it freezes tends to reject most impurities. The concentrations of these impurities will steadily increase as the water solidifies. At a certain point, when much of the water has solidified, the impurities precipitate out. When the water melts again, the impurities may not go back into solution fast enough and so they remain for you to see. If you rinse the ice cubes before allowing them to melt, the impurities are washed away and you will see less floating objects in the water (This is actually a method of purifying water). Lime, or calcium chloride, will look just like our floating particles when it drops out of water during freezing. If we agitate the ice as it melts then there may be less particles. If we heat the ice without boiling it then there may be less particles.
Hope this gives a little insight to the origin of the little buggers. Basically they are already dissolved in the water before freezing and then do not re-dissolve when thawed out due to the requirement of warmer temperature and/or length of time. Cheers.

I broke up this 4 year old thread and started a new one.
Thanks Wally for a plausible explanation. Sounds right to me. Tom