Almost everything will run on 18" radius curves, including many 4-8-4's and even some 4-8-8-4 Big Boys.
BLI brass hybrid and some of their larger 4-8-4's won't, as well as the second run of Scale Trains "J".
Most locomotives are designed with 18" as a minimum, although they can generally go a little lower if they aren't coupled to anything. I have one curve that goes 18" -> 15" -> 18", and as long as the engine isn't coupled to anything, I've had good luck with engines that require 18" minimum radius making it through. Unlucky 4 axel diesels may have trouble if there's a lot of detail though, so 15" radius is not something I would count on working.
Long story short, if you're just going to run a critter in, 15" should work. But it probably won't work for much else.