For me it depends what kind of applique I am doing.
Needle turn, where the applique piece has a raw edge plus seam allowance that I turn under as I hand sew it to the base.
Prepared applique. That is where the raw edge is already turned under and either basted or held with washable school glue I like to us a very tiny blind hem stitch by machine. That is the stitch where the machine will take 1 zig zag stitch to every 2 or 3 straight stitches depending on your machine. On my Bernina I actually have a blind hem stitch and another stitch that looks just like it except it takes less straight stitches before doing the zigzag. Some people refer to it as invisible machine applique stitch. I have also stitched prepared applique down by hand.
Fusible raw edge applique I do by machine with a blanket stitch. But I have also done satin stitch or a tiny zigzag. I just like the look of blanket stitch best.
Wool applique I do by hand with heavier thread like pearl cotton or 12 wt thread and do blanket stitch. (Some people also call this stitch a buttonhole stitch.)