Though I've learned a few techniques over the years, the one that I probably use most often for everything...except wool...is where you sew the fabric and fusible interfacing right sides together and turn. It gives a clean edge to work with, though you must be careful that the fusible doesn't show around the edges.
If I'm doing an applique project, I may use more than one technique to stitch the pieces down, depending on size of the pieces, etc. Wool applique is fun because it's all blanket stitch.
I love applique too but I do it by hand. Needleturn is my favorite because I am lazy and don't like to do much prep work. I mark my motif on the right side, finger press along the line, pin or glue (washout) to the fabric. I trim as I go so it doesn't fray. I use DMC machine embroidery thread. 100% cotton an myriad colors. I tried doing machine applique but with all the prep, it took 4 times longer than just sewing it on by hand.
I'm another needle-turner. Goes a lot quicker (for me) and it's portable - I also like to think that it's more traditional. I have done machine applique for baby & kid things, to withstand washing, and when redoing casual sweatshirt jackets.
I am a freezer paper applique-er. It seems to relax me. I can sit with hubby and watch t.v. and do the tracing and also the cutting out and gluing. Sometimes he feels like the quilt widower. Although I am a golf widow during the warmer and dryer months of the year. I don't complain because I can quilt rain or shine....If I have alot to do for gifts I can usually get one of my sons girlfriends to help out with the tracing and cutting. It is tedious work for teenagers. I have learned patience in my older years. I am amazed by the different techniques everyone favors. I am sure I will pick up some new tips for my own special way of doing things. Thanks everyone.
My original response was to the title of the discussion, which was machine applique. With the other responses and the subject being applique in general, I would have to agree with the others. My favorite would be either needle-turn or freezer paper, as I don't favor one over the other. I tend not to follow convention necessarily and may mix methods in one block.
For me, hand applique isn't as much about how I get it cut out and the seam allowances tucked under as much as it is about the stitch that I use to applique with. In my photos I've got a picture of a block with a sheep that I designed. It's all hand appliqued but uses 3 different methods of prepping the pieces. I used the fusible interfacing on the body....needle-turn on the head...and an archival glue stick for the legs. There are a total of 20 blocks to be done on that UFO, and they are all in various stages of completion. The only thing that I will change after completing that first one is that I won't use the glue stick again because it makes it too stiff to get the needle through easily.