While I am not in agreement with the whole allowed/not allowed thing, carbonated water with no added sugar should be fine. Try to go for a naturally sparkiling mineral water, that's even better.
- Seltzer water - carbonated water
- Club Soda - Seltzer Water with (possibly) other minerals like salt
- Tonic - Seltzer Water with sugar and quinine
Tonic should be treated like run of the mill sugary soda - best to avoid it. Seltzer or club soda isn't really "paleo", but it's not like you're breaking some core principle if you drink it.
We have a Soda stream. Toss the junky flavorings, but it's nice to have a nearly endless supply of sparkling water, no bottles to buy from the store all the time. When I want a flavor I add a squeeze of lemon or lime. YUM!
It may not be paleo, but I'm not going for cavewoman re-enactment, just healthy food.
If drinking seltzer water makes you enjoy the rest of your paleo food more, then drink it.
There's another use for seltzer water, though, and that's for creating an batch of water with ionized magnesium in it. Not only is it a way to get magnesium, it's good in combination with low-sodium/high-potassium V8 juice, since potassium requires magnesium to transport the potassium into cells. Yes, I know that a synthetic food like V8 isn't paleo, but I compromise for the sake of getting enough of this vital mineral into my daily diet. Add a splash of lemon juice, and I get a tasty and nutritious drink.