Corian counter tops have been a huge hit ever since they came out. People have loved getting them because of all of the different colors that they could get as well as the durability of the material. Another huge selling point about Corian is that blemishes and scratches can actually be sanded out and then the counter top refinished. Doing this isn't all the difficult, but it does take time and you need to know what you are doing.
Clean off the Corian counter top and wipe it down, just to get all of the crumbs and dirt off of the counter.
Use the green side of the green and yellow scrubbier pad on the small, minor scratches. This pad should take the scratches from the Corian countertop out pretty easily. If you have to work really hard at getting a scratch out, simply leave it and move on to the next step.
Load your hand sander with the finest grit sandpaper (1200-grit) and spray the Corian countertop with water.
Use the sander loaded with your first grit sandpaper and begin to work away the scratches, spraying while you go. You will want to switch your sanding direction every couple of passes, just to make sure that you don't leave any marks in your Corian countertop. If you are having trouble removing one of the scratches with the sandpaper you started with, move to the next roughest grit. Continue this process until the scratches are all out.
Wash the entire Corian counter top with just water and a clean towel.
Load the sander with the next finest grit sandpaper (1000) and sand the entire area with that, still spraying with water while you sand. Use a large rubbing action to blend the sanding lines. Then use the next lightest (800) and do the same thing.
Clean the Corian counter top with water again and then rub the entire area with the yellow side of the green and yellow scrubbier pad. This will give you a matte finish on your Corian countertop. If you want a gloss finish, continue to step 8.
Use a white polishing paste and finishing pads on your sander. Work this into the Corian counter top the same way that you were sanding. Large passes and spraying water as you go are key to getting a glossy finish on your Corian counter.