I've been doing a Hoopsisters quilt. It has been a very long process - about 11 months now and I'm definitely not done. They recommend (and sell) battelizer which is a batting with stabilizer built in. Great stuff and I really like it. You hoop the battlizer and do the rest of the work on it. It is not inexpensive and hooping it creates a ton of waste, so someone in the class suggested hooping the cheapest possible muslin and cut the battelizer to size. The thing I've noticed is the blocks turn out very stiff and will not make a soft snuggly quilt! I see from a previous post that the muslin shouldn't be less than a 200 thread count. Oops! Today I changed to a tear away stabilizer I get from JoAnns. It is very soft with lots of holes and I love it. The blocks are really nice and soft and are not distorting at all.
I have made a couple of soft dolls and used tearaway, but it was the more solid stiff stuff. I'll definitely go to the more porous one for the dolls. Nice and soft!