My method for fixing scratched compact discs is very simply, however I've been able to fix what most other people and products has given up on.
Caution: Your thumb can get very hot.
1. Place your right thumb on the disc surface data area near the center of the disc.
2. Start moving your finger clockwise round the center no faster than an inch in a second. Keep the distance to the center.
3. Now start pushing as hard as possible with your thumb while still moving it.
4. Expand the circle slowly while you keep pushing, so that each data area is touched at least once with maximum pressure.
5. You're done. Play/copy/rip the disc.
Sometimes is also helps moistening the thumb before, or trying again with a little moisture.
Compact disc players and computer drives differ in ability to read from scratched disc. Try several in your home to find a good drive.
It's noteworthy that CD's plays from the inside and out, and most scratches tends to be at the end of the disc.
Also, I wouldn't use toothpaste, since it contains scratching elements.
Regards from Realzneo, Denmark, Europe.