What I've observed is this: the batting and the amount of quilting both influence how stiff a quilt is.
For baby quilts, I use the thin fleece that you can get at Walmart. The quilts turn out soft and drape well - plus they are very lightweight. I generally quilt them with a medium sized quilt design because I expect them to get hard use. The medium quilt ensures that the quilt will stand up to abuse and that the quilt has a soft feel to it.
The quilts I quilt for guild charities use a poly cotton blend. It tends to be stiffer than the thin fleece. I do a medium to large meander quilting on them, mostly because of time issues. The larger meander ensures that the batting is quilted securely, but it's not a tight quilting. It meets the requirement for the most fragile bat, which is quilting no more than 4" apart.
For my own quilts that are less than 55" wide, I like to use a heavier neutral colored fleece. It's soft and the quilting shows nicely on it. Sometimes I quilt them tightly and other times I use the larger stipple. It depends on what I think will look good on the quilt and who the recipient is.
I don't make too many bed size quilts, but when I do, I use an 80/20 cotton poly blend bat or a 100% cotton bat, and quilt at least as densely as the bat requires on the label.
I used to use fusable batting, but it seemed to make the quilt stiff, so I don't use it anymore.