For machine applique, you usually only need a stabilizer if you're going to use a heavy, dense stitch like satin stitch. For straight stitch or, say, a fairly open blanket stitch, you don't need anything.
For finished-edge applique, I press or baste raw edges under, pin or baste to the backing / quilt square, and sew either by machine or by hand. I have used fusibles but mostly don't care for them.