I did not notice an issue with plastic bottles, myself, but who knows? I am unable to go to local springs and glass spring water bottles are not easy to find.
When I was in the UK, the tapwater in London was so bad that I found it killed off my appetite completely once I went fully rawpalaeo. I duly found various 5-litre plastic bottles of spring water. I found that the more natural ones, with an alkaline PH value of 7.8 to 8.4 were the best as regards taste etc. The best was a Brecon Beacons 5 litre bottle, which was so good that I found myself drinking a full bottle over 36 hours or so, sometimes. Surprisingly, I never got hyponatremia, as a result. Now, in Austria, I find the tapwater, which comes from the mountains, is fine re taste and, anyway, most local spring water bottles are too neutral or acidic.
Do NOT ever drink distilled water. I used to buy some distilled water from Planet Organic in the UK, and I found to my horror that it never quenched my thirst. Very dangerous stuff that leaches nutrients from your body etc.