There are quite a few different ways to machine applique. Harriet Hargrave has a great book out on it. (I have her older edition.)
I would have to look to make sure, but on my Bernina 1230 I think it is the blind hem stitch I use, but I mirror-image it. Whatever its name, it stitches 3 or 4 straight stitches in a row and then does a single zig-zag. I want the zigzag to go to the left, to catch the applique, so on this particular stitch I need to make it mirror-image. It's also very possible to use just a plain zig-zag stitch for the applique.
My preferred method is to use freezer paper. I trace the finished shape on to freezer paper (mirror-imaged if the shape is not symmetrical), iron it to the wrong side of the applique fabric, cut around the freezer paper with about 3/8-inch of fabric for the turnunder allowance. HH recommends using a glue stick to turn under the seam allowance. (Her book gives very detailed advice on how to do this quickly and with precision.) I prefer smearing a small amount of old-fashioned children's paste on the allowance with my fingertip (paste comes in a jar and smells like peppermint) for the turnunder, as it is faster for me and I can control the seam allowance better with it. I iron the applique piece to secure the turnunder.
For invisible machine applique, I usually use the very fine, clear nylon filament thread that HH recommends. However, some day I'd like to try silk thread in a matching color, as I think that would look even better.
For stabilizer, I simply use a heavy solution of starch on the background fabric. I mix a 1:1 solution of Sta-Flo liquid laundry starch and water, apply this to the fabric with a house painting brush, throw the fabric in the dryer, and iron with steam. The fabric comes out nice and stiff. (I do all this before cutting my background squares). If I didn't do this, I'd probably use a commercial tear-away stabilizer underneath the fabric.