I am still working on my Japanese Kamon (family crest) blocks.  Right now I am stitching around the edges of the fused applique pieces – since I never trust the fusible to be permanent.  For these pieces, I want the stitching to blend into the fabric as much as possible.  I use size 100 silk thread for this purpose, since it is very fine.  I use silk thread made by YLI and also by Superior Threads.  Your local quilt shop may carry this thread for use in hand applique, or you may have to buy it on line.  The YLI version can be purchased through Amazon (like almost anything else you could want!) at this link:  Silk Thread 100 Weight 200 Meters-
Polyester size 100 thread is also available, but I prefer the silk.  The photo below shows a couple of spools of this thread.  Note the number 100 on the top of each spool.

YLI and Superior make size 100 silk threads, good for applique and machine quilting

I iron a piece of Totally Stable (by Sulky) on the back of my block.  Totally Stable irons on like freezer paper, but is tearaway so it can be removed reasonably easily.  Other stabilizers or even newsprint could be used but would have to be pinned in place.  You may also just use starch or sizing on the back of the block to stiffen it up; for the blanket stitching I am using here, that would probably be enough to keep the fabric from puckering.  I may try that on a couple of blocks and see if I can recommend it, since it would eliminate the tearaway step.  But today I am using the Totally Stable.  Incidentally, Totally Stable comes in various sizes – 8 1/2 by 11 sheets, 8, 12, or 20 inch wide rolls, etc.  I think the 12 inch wide roll is probably the most convenient for most quilters, and you can buy it at your local quilting/craft store or here at Amazon: Sulky 12-Inch by 12-Yard Totally Stable Iron-On Tear-Away Stabilizer Roll

After the Totally Stable is in place, thread your machine with size 100 silk in both the top and bobbin.  Check the tension – I like a setting of about 2 for my Bernina, but your machine may be different.  The bobbin thread should not show on the top if the tension is correct.

I set the stitch length to 1.5 and the stitch width to about 0.7.  I am using the built in blanket stitch (stitch 329 on my Bernina 180).  I like a pretty narrow stitch so it doesn’t show much.  The resulting stitch shows below on the lower leaf of this block; the other leaf hasn’t been stitched yet.  In this closeup view you can see the stitching.  From a little further away, it is nearly invisible.

Lower leaf has been machine stitch appliqued (blanket stitch with silk thread)

I think this method is a good compromise.  The stitching is inconspicuous and it is fairly fast, so I will make reasonable progress.  If I try a block without the Totally Stable, I will report here how that went.