I starch when I am going to be piecing, especially a fabric that feels lightweight. The starch helps stabilize the fabric and keeps it from shifting when I am piecing. I also starch when making berries. I use the 'Perfect Circles' mylar circles to make them and it takes a bit of starch to get them perfect. And anything on the bias usually gets starched to death from me.
I do not use starch when I am appliquing! That said, there are times when starch can come in handy on applique. If you notice the fabric is starting to fray at a point or v, dipping a small brush in some starch and brushing it against the fraying area will help keep it from fraying further. Of course, a little smidge of Fray Check or some of Roxanne's Basting Glue will do the same thing. I think it just depends on what you have on hand and what's easiest at the time.
For those with Celiac Disease or who are allergic to wheat, you do not want to use commercial spray starches unless they specifically state they are gluten-free. You can make your own starch the hard way, by boiling up some rice in a little bit of water. You'll know it's done when the rice releases its starch into the water, turning it a milky white. It's a bit of work though and I'm lazy. If you don't want to go to the trouble (or waste the rice), you can use potato starch powder or even corn starch powder. I'm allergic to corn so that doesn't work for me but the potato starch does just fine. I imagine tapioca starch would work just as well too. Just thought I'd throw that out there.