I am a novice when it comes to applique. I am making placemats and wonder if I should put some type of fusible batting or backing on my piece I am going to applique. I am making dresdan plates to go on the mats. If so, what should I use. Thank you.

I like to use freezer paper, iron it to the fabric, then cut around it with an extra 1/4". Then I use a glue stick around the 1/4", fold the 1/4" under and iron it. Stays like a champ! Then if I need my piece to stay put, I also use a glue stick on the wrong side of the piece, put it in place and iron it. Nice temporary hold without the cost.

It's a good idea to stabilize the background fabric in some way when doing applique.

Instead of using a stabilizer, I like to heavily starch the background fabric. For this I use a 1:1 solution of Sta-Flo liquid laundry starch and water. This leaves the fabric with about the stiffness of cardstock, so it is very stable.

Are you planning on doing raw edge fusible, or a turned method, piecing the dresdens?
Knowing this, will help us answer you better :D:D:D

I was and did raw edging and machined around the outside to put it in place on the mat. I was just thinking I need something behind it to make it firmer and no stretching.

Don't you need a stabilizer with both?

Not sure I understand. You mean you have completed the applique and are just looking for a way to finish the mat? You don't necessarily need batting or fusible if that's the case. You do probably want to use a backing fabric and binding. You could add a fusible to make the mat stiffer or a thin batting if you want to quilt around the design.

For future reference for myself. Should I stablize the plate before putting it on the mat?

It is more important to stabilize the background fabric than the plate. It is the background fabric that is more likely to distort.