I don't know, I have a black one and I can't see anything through it. It might be a little thin in white I suppose, but TE doesn't carry them in white. I guess it would depend on how tight it fits, too; as I mentioned my cup size is pretty small. I don't know why you couldn't wear it on its own (in situations where you'd wear other sports bras on their own); there are no hooks or strap adjusters or anything that screams "underwear."

If you're wondering whether your nipples will protrude, mine do no matter what I wear so I can't really help you there.

I love the little saying on the inside of the band. You will have to buy one to know what it says.