Well, I make a lot of my and my children's clothes out of cotton or cotton blends, so a lot of my clothing scraps are quilting cottons. I've never tried to use knits in a quilt, not sure they're stable enough. For a backing though, it might work, because the woven front would stabilize it.
Hardcore quilters will say never use poly blends, but I do quite often, it doesn't make that much difference, and in fact I've used rayons and even 100% polyester wovens. But the 100% poly has raveled a bit with washing, so I'm not sure that was a good idea. But it was also my first quilt and so it is possible the seam allowances were on the scanty side.
I've also used flannels in quilts, and have a bag of flannel scraps I intend to make a child's quilt out of.
Oh yeah, some people do use knits in quilts, for tee-shirt quilts, but I've never done one of those, and you have to do some things differently, I think.
Edited to add: If you want to make backing from stash, you can also piece smaller pieces together for the backing. I've done that several times, and the effect can be really nice.