I would hate to harm the soft material. What cleans it best without scratching or damaging?

If soap and water won't do it, I certainly would not apply or rub it with anything but a soft flannel cloth. Anything more abasive would undoubtedly leave scratches on the plastic. Maybe someone with an open car will chime it on this.

Meguires sells a plastic window polish that works very well. I've seen detail shops use a small foam buffing wheel to make plastic windows on convertibles crystal clear.

I use a spray on wax and a soft cloth. Seems to work really well. I just spray very lightly ie. a fine mist.

I have used Goof-Off paint remover with great success to clean off years of grime on my 1924 Touring rear plastic windows. Simply rub on with a paper towel and follow with a soft cloth. It does take a few applications (rubbings) as it acts similar to when you clean glass with acetone to get off sticky adhesives from stickers/decals.

If your window has a sticky or gooey film on one or both sides, use WD-40 to remove it.

WD-40 is gentle and has no agressive solvents such as acetone or lacquer thinner. I've used it on top windows that were impossible to see through and when done, the window looked new.

If the window is scratched, I use a product called Novus. It comes in different strengths and is formulated to clean and polish plastics of all types. Novus #1 is a thin liquid in a pump-spray bottle and is used mainly to clean and lightly polish plastic. The numbers go up from there and they increase in polishing effectiveness. Novus #1 is the only one offered in a spray. All the others are in paste form.

Goof-Off? That's kind'a scary any plastic we've put goof-off on it ate it up... even a pump up sprayer hose. Be careful with any paint remover you might find yourself back into a mild or full blown restoration project.

No problem at all with Goof-Off cleaning my plastic windows...just don't get any on the paint!

I have used it multiple times with no ill effects [unless I breathe in the fumes!!]