Moonlight can produce rainbows, in the same way that sunlight does. However not many people have seen a moonbow, so count yourself lucky if you do! Moonbows have faint bands of colour. However it is difficult to see these colours because of the low intensity of moonlight, so Moonbows appear white.
The fine droplets of fog can also create a bow like a rainbow. However fogbows are white. You can see the concentrated light produced by the water droplets, but the colours do not separate.
The evenly small droplets of fog create large interference patterns. These interference patterns are so big that all the different colours overlap, recombining to make white light. The result is a white bow.
Normally interference can only be seen in the supernumerary bows along the inside of a rainbow.