Silver does not rust because it is not an iron-based metal. However, some silver-plated objects may rust if they have some type of iron in the plate, or if the surface underneath them is susceptible to rust. In most cases, any "rust" that appears on a silver or a silver-plated surface is actually tarnish. The best way to deal with this type of stain is to remove it using processes and cleaning agents that will work on tarnish and rust without damaging silver or silver plate.
Use 1 tbsp. of baking soda and a damp rag to buff off tarnish and rust. Dampen the rag, then sprinkle the baking soda on the rag. Rub the stained spots with firm pressure and a circular motion, as if you were waxing a car. As you clean, the stains should come off and the silver plate will shine. The baking soda will not scratch your silver plate.
Tackle ornate silver plate items with hot water and baking soda. Place a sheet of aluminum foil flat on the bottom of a pan. Fill the pan 3 inches full of water and add 1 tsp. of baking soda and 1 tsp. of salt before bringing the entire thing to a boil. Once it is boiling, add your "rusty" silver plate items and boil them for 3 minutes. The rust will dissolve in the hot mixture, as will tarnish.
Remove any remaining rust with toothpaste. Cover the entire silver item with toothpaste. Hold it under hot running water and rub it until the toothpaste foams. You can use a soft baby toothbrush to work rust and stains out of crevices in the pattern.
Wash your silver plate item off with hot running water. Make sure all residue from your cleaning agents is gone. At this point, you will be able to see whether your silver plate has flaked off, exposing the surface underneath (which should now also be clear of rust, thanks to your toothpaste treatment) or whether you had a simple tarnish issue.
Buff the silver plate pieces dry. Use a clean, dry cleaning rag. Gentle, firm rubbing will make your silver plate shine and remove any smudges that might dull its gleam.