For hand quilting, I have used many of the cotton batts and have not been happy with the results.  They often beard, which is when the fibers come through to the top while quilting.  They also make a mess as you are quilting, because the batting fibers that are not between the layers, but around the edges, tend to shred and get on the quilt top.  Another draw back is cotton batting can sometimes be more dense and therefore a bit more difficult to hand quilt through. 

Cotton/poly blends work very well! And my favorite is Hobb's Heirloom 80/20.  The cotton is thin and even and the poly portion allows you to needle easily through the layers. 

100% polyesters are great for wall hangings, quilted clothing and for getting tiny quilting stitches. 

I use Hobb's Wool Batting for most of my quilts.  I love this 100% wool batting!!!! Don't be scared off by wool. This batt washes very well and is so easy to quilt through.  It has a slightly higher loft, but that only enhances your quilting stitches.

A word on washing and shrinking battings.  I pre-"wash" all my batts before I put the layers of my quilt together.  I am not looking for an antiqued look for my quilts so I want all the shrinkage out of the batt, the same reason I wash all my fabrics.  You can read more about this in my blog post regarding pre-washing fabrics.