How do I make a Noise texture (or any texture) only appear in places where Musgrave is white/black?
I tried adding them with math node but this gives monochrome image.
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There are lots of ways to do this, but to open up mixing options, you could do it via the colors of the textures, via a Mix RGB node set to one mode or another -
If you just want to use the Musgrave Textrue as a mask, (not actually see it,) then you could connect it to the Mix RGB node Fac to Fac, and put whatever other texture/color you want into the top color slot.
Instead of using the Math node for adding, you should use the Math node with Multiply. So for example then the areas where your Musgrave texture is 0 will remove the influence of the Noise texture there.
If you use colorful noise you could have a color mix node with that noise and white (or whatever color you want when the musgrave is 0) rbg color as inputs, and the mix factor would be controlled by the musgrave texture value