Different size raindrops make different rainbows. Fog droplets make a colorless fog bow but frozen fog make unique colorful rainbows such as the one below.
https://www.metoffice.gov.uk/learning/clouds/fog/freezing-fog
The picture below is an ice fog rain bow.
So to conclude Ice equals color and distinct rainbow pattern when it comes to fog.
Diamond dust is what frozen fog is called. https://saebloom.wordpress.com/2016/12/13/diamondsnow/
All the patterns are generated as sunlight (or moonlight) is reflected and refracted in flat six-sided water ice crystals in Earth's atmosphere. https://apod.nasa.gov/apod/ap181221.html
I found this picture of High clouds refracting light. https://www.dkn.tv/tin-giai-tri/nhung-vang-hao-quang-ky-la-tren-bau-troi.html I could get my translator to work but just looking at the picture the sun is past the horizon of the camera but not the clouds and clouds that high is frozen creating a Fire Rainbow.
https://physics.stackexchange.com/questions/289279/why-is-each-snow-flake-different
To add: You could use a telescope and angled white laser or focused light to synthesis a bow to see if it refracts differently between ice/color or liquid/no color and the pattern it makes.
Added After Edit:
http://contrailscience.com/aerodynamic-and-rainbow-contrails/
If the contrail has refraction or rainbow it is ice. The pilot can turn the plane to the angle needed to see the rainbow from the cockpit.
Rain bow at night with spot light. https://atoptics.wordpress.com/2017/02/05/binary-double-rainbows/
If this was ice or fog rather than rain the rainbow will be different in pattern and color. I am interested to try this with the whitest of light or maybe a white laser which does exist.
Last edit: I believe the sunlight is refracted inside the ice and is warmed because of that there must be a equilibrium between the cold air and ice to stay frozen during the day but at night the ice fog could be warmer. I have not considered if the friction between the air and ice as a factor.