We have recently moved, and I've just arranged my twillies, scarves, and shawls in their new home this week. It's a large chest of drawers with working locks. I've dedicated the top drawers to them, and have folded the majority of my collection so that they have an edge up for ease of selection. Fortunately I have enough to fill out two of the drawers this way, and a third contains my small assortment of twillies and scarf rings. One drawer is still left for room to grow.

I share these humbly, as I'm not certain if this is an ideal storage configuration, though it pleases me at the moment. I'm learning from our fellow scarfies here as I peruse threads that I still have so much to learn!
**editing to add: I also store them digitally by keeping a photo album with each piece in a flat lay, sorted by predominant color. This lets me visually browse when I'm planning my outfits so that I don't have to disturb my silken treasures too much between wears.