Sorry to say I have no hands on experience, but you’d need a below-freezing microscope, which could be done with an ordinary light microscope once it’s slowly cooled down. Cooling it too fast might be risky for the optics. I’d try catching falling flakes on clean glass slides pre-chilled well below the freezing point.
There will be a problem with fogging of the eyepieces when you bring your warm, moist eyes next to them. There are anti-fog coatings and possibly (?) heated oculars. Telescope people would know about this.
I recall a long-ago Scientific American article (amateur scientist column) on this topic. There are lots of beautiful photomicrographs of snowflakes published in books and no doubt on the internet as well. I’m sure there’s a body of how-to literature on this—somewhere. Google around.
Might I suggest a video microscope? You can get cheap ones (well under $100) online up to 60x power. Snap a photo of the snowflake then examine it in great detail later. Hi-res will cost $$. It’s also possible to retrofit a good optical microscope with video camera as an image capture device.
Lighting is a separate issue. Back- or side-lighting might be necessary to bring out certain structural details. Of course lighting usually generates heat which will melt your subject. Consider remote light sources connected to the “snowflake stage” by fiber-optic cables (like surgeons use on their heads), or tight-beam spotlights located at a distance.