How deep are the scratches? I have heard that toothpaste (of all things) and buffing might work on glass and plastic of the scratches are shallow.
http://www.ehow.com/video_4872903_remove-scratches-from-plexiglas.html This worked for some minor surface scratches I had... not to sure about deeper scratches though
I've heard this as well I think it's because it mildly abrasive , a bit like cutting compound, but I would'nt recommend brushing you teeth with cutting compound
I have used Autosol for taking out scratches of plexiglass windows with good results, but as anything like this test it somewhere first
Tcut, brasso, Meguiars PlastX and Novus polish are all things you'll find in my consumables cupboard for repairing or restoring plastic that has become scored or faded.
On our Aircraft perspex we use a system called Micro-Mesh. A variety of abrasive rolls ranging in grades from 1500-12000 (Standard Wet & Dry is between 150-600 grit). The higher the number, the finer the grade. A lot of elbow grease involved but excellent results can be achieved. Just be aware you are actually removing the material so it depends on how deep the scratch/groove is before you achieve a smooth finish. Start with the lowest number till it feels smooth then move up to the next highest number, rinse and repeat until it is perfectly clear again. At each level expand outward slightly from the damaged area to avoid "dishing". Oh and dremels should be avoided - hand work only please. Cheaper than replacing a whole perspex panel... Enjoy!