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Leftover foodstuffs can cover up stains, shielding them from your cleaning efforts. Rinse and dry the dishes before continuing. To remove deep stains from your dishes, you'll need to introduce a solvent to dissolve the stain so that it can be washed away. Baking soda is a popular and convenient option and is significantly less caustic than industrial solvents. Start with a tablespoon of baking soda, and add just enough water or white vinegar to make a sticky paste. Using a clean dish rag or scouring pad, scrub the stains vigorously with the solvent paste, then rinse. Lemon juice is another common household product which can function as a mild solvent, and is a good substitute for white vinegar. If a baking soda paste has proven ineffective, you may need to allow the solvents more time to deeply penetrate any stains. In enough hot water to completely submerge your dish, dissolve roughly 1 tablespoon (15ml) of baking soda and 1 tbsp of white vinegar per cup of water, then allow the stained dishes to soak in the solution for 1-2 hours. If the stains have faded somewhat, but not completely, you may repeat the baking soda and vinegar soak. If the soak has not been effective at removing the stains, you'll need to consider more heavy-duty solvent options. If common household solvents have proven ineffective, you may need a more potent solvent to remove the stains on your dishes. Many brands and varieties are available. Whenever working with chemicals, be sure to follow any instructions on the product's package closely. Work in a well-ventilated area, and avoid exposure to any caustic chemicals by wearing rubber gloves. Rinse your dishes thoroughly after applying the solvent to ensure that none of it remains on the dish. Not all commercially available heavy-duty dish cleaners are appropriate for all types of dish. Read the packaging carefully, and familiarize yourself with your options before purchasing. Chlorine bleach, or any product containing it, should not be used on ceramics or glazed china, as it may interact with, and ruin, the glaze. Oxygen bleach (usually used for laundry) can be used instead. Mix oxygen bleach powder in hot water, allow it to cool to room temperature, and allow your dishes to soak. This will remove most stains, even the crack-like "crazing lines" which may begin appearing as ceramics age. A common alternative for removing stains on china is 20% hydrogen peroxide solution, which is available in most pharmacies. Simply apply the solution to the affected area of the dish, then wash and rinse thoroughly.

Summary:
Wash dishes thoroughly. Spot-clean stained dishes using a baking soda paste. Use a soaking solution of baking soda and vinegar to remove deeper stains. Rinse the dishes, and evaluate any remaining stains. Try a more powerful solvent. Use care when bleaching ceramics.