For machine applique, especially if you are satin stitching (but also if the appliques are quite large), it is best to stabilize the background fabric. Instead of using stabilizer, however, I heavily starch the background fabric (think 1:1 Sta-Flo liquid starch and water). Heavy starch stabilizes the background fabric enough that it won't stretch or distort while you are machine sewing around the applique.
Fusible stabilizes the applique; however, the background fabric also benefits from being stabilized. If you have not washed the background fabric, sometimes there is enough stabilizer in the fabric from the factory. However, if you have washed the background fabric, typically it is more limp than when it came right off the bolt. Starching the fabric will restore and even add stability.
If you think about it, when you are machine appliqueing, there are times when you are sewing on the bias of the background fabric. The bias is the most likely to stretch. When the background fabric is heavily starched, the bias won't stretch.
The fusible with the least stiffness is Misty Fuse. However, you need to Google tips on using it first. It does not come with a paper backing like other fusibles, for example. What I do is iron it between sheets of parchment paper, peel off one of the parchment paper sheets while it is all still warm, and leave the other one as the paper backing. I go to this extra work because I like the fact that it does not stiffen the appliques at all; it is as if there is no fusible at all there. With experimentation I found that the old Steam-a-Seam was the best of the paper fusibles; however, I have heard the new Steam-a-Seam is not as soft. Years ago I tried Wonder Under, but it was way too stiff for me even after washing.