I use tear-away if the project has a dense design or fusible no-show mesh for less dense designs when removing the stabilizer might damage the bobbin threads.
I like the no-show mesh cut away. It's soft and flexible and I just leave in the entire piece. I have stuffed many items with it and had great results. I avoid pulling stitches because there is no tearing and no misshaping of fleece. It works for me.
Tear away - and get rid of as much as possible before you turn the pieces! (Especially around the seams.)
In addition to all mentioned below, I have had good results using a good quality batting; one that is firm and stable.