Does anyone know what the best adhesive for hanging plexiglass mirror

I would try command strips or buy mirror brackets that fit around the edges. Home Depot and Lowes both carry these items.

There is a product sold at Home Depot called MirrEdge. It frames the mirror and then you can use something like Liquid Nails adhesive.

If you are putting it up permanently, Liquid Nail will do the job. Otherwise, I believe there is a new product out at the big box stores made by Scotch that is available. You could also use velcro.

The Command strips work well to a point, if your estimate the weight limit, be prepared for disaster. I lost a collectable plate (framed) to Command strips.

Personally I would tend more toward the mirror brackets.