I have to lovely cats too - they are both boys, and they've destroyed more than one of my dresses etc. I've found the very best option possible is to gently fold them and dedicate a drawer to them. Shake them out, and they are good to go. If that's not your thing, I'd highly suggest using a garment bag that is shaped like a huge cube and hang it up in your closet. The most delicate things(dresses that the kitties might snag that could not be fixed and delicate items like netting that are easily ripped would go in). They have some great options under $30 at place like Target and Walmart. Look for something that is canvas and not mesh netting since the cats can cut right through that! Also, canvas storage boxes work FANTASTIC. The ones that are intended to slide under the bed, I just put it in the top of the closet - a very safe, safe place from the cats and a huge plus is how light it will be since your crinolines will be what's inside. The boxes run anywhere from $7-$40 depending on where you're shopping. I've found some for cheap at places like Ross.
I hope that helped. :]
Unfortunately, I've tried those options and they aren't working for me. I have one of those cube hang-y things from Ikea. The crinolines just don't fit very well in them. And when I've put them in boxes, they just don't look very good after I've taken them out. I don't have much space in my apt for an another zippered closet box. I guess I'll just have to keep looking. Sigh.
I hang my crinolines from a skirt hanger in my closet. Towards the front, where they have the room to be full. They seem to be ok, and there has been no type of damage to having them this way in the closet. Occassionally, when I have an extra dry cleaning bag, I'll hang that over the crinoline. Hope that helps.