Generally, a damp microfiber cloth used with a standard household kitchen spray should remove most stains.
For most residues, all you need to clean your Corian® Solid Surface countertops is warm soapy water, ammonia-based household cleaner, or a dedicated countertop cleaner such as Bar Keepers Friend or Cif. Always clean using a circular motion.
Hard water areas can cause a film to build up on countertops if water is left to dry on the surface. This film will dull the countertop surface, making the finish appear blotchy and uneven. To prevent film build-up, it is best to dry surfaces using a microfibre cloth after spills and cleaning.
This cleaning video shows how quick and efficient it can be to clean a Corian® Solid Surface.