I do not cut

I seldom trim away the fabric underneath - unless the appliques are bulky- or I for some reason its large & I want the fabric that's under- then I've been known to- but generally I like my quilts to be fairly heavy/substantial anyway- so I leave the (foundation)

I don't cut it away because to me the appliques then look like a smile looks with teeth missing...sort of caved in.
If I can't quilt through 2 pieces of fabric I'm pretty weak. :)

I can't answer for fused applique, never done it.

Thanks everyone - that really helps me know what to do. My project is a small wall quilt - won't be too heavy. I'll be free motion quilting it so thickness is also not going to be a problem. (I used invisible machine applique by the way! Don't trust my needle turn skills and I'm not that patient!)

I do cut away excess backing fabric on applique-- for two reasons. When I handquilt, it's a lot easier -- that's not 2 layers of fabric, that's potentially 4 plus batting. It's hard to get nice even, small stitches through that much bulk.

The second reason to trim away backing is to allow the color of the applique to ring more true -- as an example, I had yellow flowers appliqued all over a dark grey background. By clipping the backing away, my yellow fabrics look their best and stand out far more than if I had a layer of gray behind them. I do had work needle turn applique, and even though I clip pretty aggressively and wash my quilts a lot, I've never had one fray or come apart. If you quilt fairly heavily the quilt is well stabilized and I really don't see that being a problem.

I don't cut away the backing, I think that it weakens the quilt. The first and only time I did cut the backing away, the appliqué was wavy and I didn't like the way it looked. I couldn't see any difference with it cut away either.
Sue

I cut away my backing if I can see the backing fabric through my appliqué. Or if there are stems (for leaves or flowers) that run behind the appliqué.

I don't cut away. Even with fusible I leave it to give more body to the applique. I have never found it too stiff or heavy.

What little I have done on my 4 x 4 frame, I cut away. BTW, I use dryer sheets behind… They hold up & smell good...

But what if it's like 4 or 6 pieces?