I stitch points much like Poly above. One thing that I do do when changing directions is to take one extra stitch at the tip of the point or at a deep "V". Once you have taken this extra stitch, remove your pins, peel the side back exposing what just stitched and trim very close to the stitching. Add your pins. Trim the very tip slightly. Using your dampened pick, cause a 45 degree angle. When it is in place, stuff and stuff with your needle, and then stitch the second side. Sometimes, I stuff more than necessary and then use the tip of my needle to coax the extra out again to my markings.
Incidentally, I mark the top of my piece with the white marker by Clover and use it as a guide to the shape of applique. (The marks disappear with heat or water.) Also, my seam allowance is about 1/8". This narrow allowance is scarey, but can be done. AND, use a very fine needle. I use a straw or milliners needle, #9. Thin needles do not make big holes thus separating the fabric weave causing the freying.
Hope this helps.