Distilled water tastes like shit, despite what water distiller salesmen would tell you. The only thing it's good for is topping off a radiator.

I have at least a dozen conspiracies to believe in before saying that in general, tap water is bad for you. There are many places in the world I wouldn't drink tap water, but where I live, (Ontario Canada) I have no problems with the municipal supply. I drink much more water than most people I know - seeing as I don't drink any type of sodas. Apart from water there's wine and some beer every now and then. But since those aren't daily, water is "it" over 90% of the time.

Right now we use a Brita filter jug, but that's mostly as a convenient vessel to hold the water in the fridge rather than for filtration purposes. When I get the new fridge I'll be using its built-in filter and don't plan to install anything under the sink nor whole-home.

If you're concerned about the purity of your water you should be very concerned about filtering whole-home because you also absorb a lot of contaminants through your skin when in the shower and the bath.