It says a lot when we can all assume that the public supply is safe to drink, even if sometimes unpalatable. In most parts of the world you are taking at least a risk when you drink unboiled or unsterilised water- note the current cholera outbreak in Zimbabwe, where even hospitals are without a clean supply. That sadly doesn't mean that bottled water is necessarily a safer option even there- the bottling plant (which will be local, water's hellish expensive to transport) might have poor standards of sterilisation and hygiene.

I don't know what allows me to have chlorine in the form of sodium chloride but not to drink water with chlorine or flouride in it.

That's because the sodium has taken the sting out of the chlorine, and vice- versa. the chlorine in tapwater is there to kill the bugs, so they leave the sting in it. They never put fluorine in tapwater, it's horrible stuff to handle. They put a fluoride (usually sodium fluoride) in some drinking water supplies as a preventative against dental decay.

Water distilled in a properly maintained plant is safe- though probably pretty tasteless- and it's only really feasible where copious solar or geothermal energy is available.

I'm lucky- our own local supply (from the River Derwent, it's only been through 3 or 4 small-to-medium towns before it gets to us) usually tastes really good. It's also soft as a baby's bum.