In a pinch I have used tap. I have used bottled water, too, but not mineral water. I don't sell and I make really small batches, so it's not a big deal if I get DOS on a bar-- I just cut it off.
Most often, though, I use distilled to avoid any issues.
I use vinegar instead of water in almost all my batches. Sometimes I use AVJ. So those are also acceptable alternatives to distilled water, although you will want to account for the extra lye needed with vinegar.
Also, for my first master-batched lye, I used distilled water, because I’ve read here that vinegar thickens the lye solution if left for long.