Threads on this come up periodically on 2+2. If I do say so, there's usually some serious misinformation tossed about. I've been picky about my water for years, so have tried to keep on top of the issue. Still, the answer, IMO, is a bit muddled. A few points:
1) Those who insist tap water is generally purer than bottled are only right if they're talking about some instances of spring water which has been left unprocessed. Bottled water consisting of tap water purified by reverse osmosis or distillation is obviously purer than unprocessed tap water.
Moreover, the tap water in many US cities does contain some toxins you really shouldn't want to drink. Arsenic is a good example. Some cites, like Albuequerque, have very high levels of arsenic in their tap water:
http://www.nrdc.org/water/drinking/u...lbuquerque.pdf
So there can be good reason to want to drink something purer than the straight tap water of your city.
2) Muddying the waters though (no pun) is the issue of the environmental impact of bottled water. It's certainly an important consideration.
3) Also, complicating the issue is that there is evidence that drinking water stored in plastic may not be good for you. Many kinds of plastics leach substances such as BPA, thought to interfere with the endocrine system:
http://www.ourstolenfuture.org/
So bottled water may have its own problems.
Incidentally, Nalgene sports water bottles may be among the worst offenders in this regard. There are stainless steel and other alternatives.
4) You can install a home reverse osmosis system, or a home distiller. If you don't care about the wasted water in the former or the extra power usage in the latter it could be the way to go. (But those are environmental considerations too.) Distillation probably does the slightly better job of purifying water. Both remove arsenic and almost all other harmful sustances.
5) One good way to go, is to live in a town with exceptionally pure tap water. I'm lucky enough to find myself in this situation now. We use an under-the-sink filter as well, just to get out some basics like residual chlorine.
If we lived elsewhere, I'd seriously consider RO or distillation.