I have 3 pop up drying things from container store. They are round with netting and fold up small. I rarely need more than 3 though. Sometimes I hang lightweight things and pants.

I do laundry once a week. I do a separate hand wash cycle load for sweaters, etc

I have a cheap Ikea rack that I use to dry clothes on.
Every once in a while I wash a couple of woolen sweaters, put a towel on the rack and spread the sweaters flat to dry on the towel. No indentations from the rods.
I wash sweaters rarely - I shake them out and brush them after use. Woolens are quite resistant to dirt, and we never wear them next to our skin.

On the ironing board, dryers, folding table. No special racks.

I put them in the dryer on extra low, or lay them out on the carpet in a corner. But we live in a dry climate.

I hang them on a drying rack. I have 2 large ones and I have tons of. Items that I don't put in the dryer and this seems to hold them all

I dry clean my sweaters. But I can get a few wears from them.

I lay them on the bed. Problem is the bed stays wet for a day after.

How often are you doing laundry that you have 10-12 sweaters in your load? What is the fiber content of your sweaters (you may not need to flat-dry all of them), and how many times are you wearing them between washes?

To answer your question, I will have maybe 2-3 items to lay flat every other load. You want to lay items flat to dry if they get very heavy when wet and will stretch too much if they are hanging over a railing. You can also put them on towels on the floor, and then hang them once they are nearly dry (and when there's not so much water in them that they are heavy).

We have this. Works quite well. http://www.amazon.com/gp/aw/d/B0060JIP98/ref=mp_s_a_1_1?qid=1414907266&sr=8-1&pi=SY200_QL40

I either lay them on the top of the washer/dryer or the ironing board.

I just put them on the rods of the frying rack. I shake them out first, and iron them afterwards. I don't put anything in the dryer, and I dream of having someone come to my house just to iron.

Check the "Solutions" catalog. They have some for less than that. That said, it's annoying to me that those things don't fold down easily, so we got rid of them. Our house is too small for stuff like that. I use our IKEA drying rack, which folds flat. It's the big metal one. You can dry two sweaters at a time.

On my furniture. I ain't fancy.

I have folding tables in my basement (where the laundry is) and I just lay them on towels on the tables.

I want to start laying my sweaters out flat to dry, but the flat racks I've found are expensive (around $100 to stack as many as I'd need) and bulky/noncollapsible. Every time I do laundry, I have 10-12 items I'd like to lay flat. Do other people really just have 1 or 2 sweaters per load? I've seen hanging lay-flat systems, but they only have at most 3 tiers.

I've been draping them over the rods of traditional drying racks, but they wind up wrinkled, stretched out, and indented from the rod.

Hoping someone has an ingenious solution!