When I make a quilt, I study the pattern first and then cut the large pieces out first and then the smaller ones. Before I even do that. I tear off the selvages first, always. I will then tear all the borders off too, as I like my borders to be on the straight of grain always.
The selvages are very strong. I roll them in a ball and put them in a special bag. I use these to tie my quilts onto rollers. All my quilts are rolled on cardboard rollers. The kind fabric comes on...The selvages are used to tie many things, I find a good use for them, even hubby uses them and they must never be include in your quilt. Tear the selvages of the backing too.
I never throw my scraps away. I use the smallest of pieces. I just joing them up and create something. The trick is to make a S&C quilt, ever so often so the scraps don't pile up. I love making scrap quilts.