Just to add to Dave's wonderful post. The latest craze is "raw" water, which is unfiltered and untreated surface spring water or simply runoff. It even comes with an expiration date--it will turn green after that. You are taking your life in your hands with that. It might be safe as a vodka mixer, but even then...
Just like cameras, there is no "best." I will separate bottled water into four categories:
Spring Water: spring water is ground water that flows naturally to the Earth's surface / on the ground. It has some hardness minerals in it like calcium and magnesium, plus sodium, carbonates, and other elements naturally found in ground water. Spring water is usually preferred by folks who are more "taste aware," so it is generally considered the best tasting. Spring water always fascinated me because the only difference between well water and spring water is that well water has to be pumped; spring water flows naturally out of the ground. That's really the only difference, yet well water gets a bad rap while spring water is coveted. Marketing is a wonderful thing, eh?
That's not quite fair, because spring water that is bottled comes from select springs that don't have some of the troublesome contaminants like iron, manganese or sulfur. Those troublesome contaminants taste and smell bad. Some spring waters have those contaminants just like some well waters, but no one would bother bottling and selling water from those springs because, well, no one would buy it! When you build a house and dig a well, you are kinda stuck with whatever water is in the water table under the property. That's why well water gets a bad rap.
Anyway, spring water is usually considered the best tasting, but from a "safety" standpoint it is no better than city water. In fact, city water is monitored, tested and regulated more than bottled water. Anyone who thinks spring water is generally "safer" than city water is misled. There are always exceptions...
Purified Water: usually this is city water run through a combination of water treatment unit operations like backwashing filters, softening, reverse osmosis (RO) nano filtration (NF) and ultraviolet light (UV). If "safety" is your main concern, and/or you want the lowest level of contaminants in your water, then this is the bottled water I would recommend. Dasani (Coke) and Aquafina (Pepsi) both fall into this category, although Dasani adds back some minerals. I constantly put the word safety in quotes because here in the US, most water is quite safe. Even the lead fiasco in Flint wasn't that bad compared to some parts of the world where people die every day from waterborne illnesses. [please don't read that to mean our water infrastructure is fine; it needs serious attention and funding]
Distilled Water: distilled water is boiled water. This is what most people put in their irons to keep them clean. Some people say you shouldn't drink distilled water, thinking it will suck out the minerals from your body. It really is fine, just a waste of money to drink it. RO treated water (purified water) is pretty close to distilled water quality (and better in some respects). No point in drinking distilled water, but it really won't harm you.
Flavored Water: Starts out as purified water or spring water, then sugar and/or other stuff is added. Yeah, whatever...
It really comes down to what water you think tastes the best; the world is your oyster! They are all essentially just as "safe"; the safety differences (including city water) are marginal at best, and "safe" relative to what and which contaminants? As far as nutrients go, even if you drink the recommended 8 glasses of water a day, the amount of nutrients you get from water is inconsequential compared to food or one glass of milk. Literally milligrams instead of grams. Drinking one type of water over another for the nutrient benefit is like hand-holding a 10-minute exposure with IBIS activated. I just isn't going to matter.