Here are the results so far:
3M Super 77 Spray - I had used this with very good results on a module. That backdrop was a different brand and material. Glad I tried it out on a scrap from this back drop first as the results were awful this time. As can be seen in the photo, the adhesive reacted with the vinyl backing of the backdrop causing bulges and wrinkles.
Elmers Glue-All - The common white glue that many of us use for cork road beds, track, and scenery. I had also used this before on a backdrop on an old layout with good results. That back drop was paper backed. This seems to work well with this Backdrop Junction back drop with no interaction with the material and any that oozed out on the edges cleaned up easily and left no stain.
Roberts Vinyl Tile Adhesive - This was left over from installing the floor in the train room. While it holds well and gives a guy a little time to work with, anything oozing out on the edges easily stains the back drop.