A grimy case is easy to fix. For silicone and clear cases, remove the phone, then wash the case with warm water and a drop of mild soap, using a soft toothbrush for stubborn spots. Rinse and let it fully air-dry before putting your phone back — trapped moisture is the enemy.
For clear cases with yellowing, a paste of baking soda and water gently buffed on can lift discolouration. Leather cases should never be soaked; wipe with a barely-damp cloth and use a leather conditioner occasionally. Avoid harsh chemicals, alcohol and bleach on any case, as they break down the material and cause cracks. A quick monthly clean keeps your case fresh and your phone hygienic.