We keep glass bottles of filtered tap water, fizzed up with a soda stream, in the fridge. Really nice and not too expensive, though the dentist tells me I should really have flat water most of the time.
When I was at uni one of the projects every first year of my course did was testing a bunch of waters. Generally speaking, tap water was significantly healthier than the bottled waters, which have a bunch of contaminants that don't tend to be in tap water, like arsenic and heavy metals, pharmaceuticals, etc.
Nowadays, the majority of bottled water you can buy, especially the cheaper stuff, is just tap water that's been bottled, but it's worse for you because it sits in plastic bottles, often exposed to the sun and heat, and that means plastics and chemicals from the plastic leach into the water.
I'm okay drinking water straight out of the tap but I do prefer it cold and I can taste the chlorine in it which I don't particularly like, so that's why we filter. But I understand just having it sit in a glass jug for a few hours will get rid of most of the chlorine.