This one is kind of icky/smelly...

Soak it in a little warm water and peroxide and then brush with toothpaste. It's okay to brush it with toothpaste each time you take it out.

Is it plastic, plasticine, or rubber?

If it's not rubber, you can both toothbrush it (soft bristle brush) and use a denture cleaner on it. However, the stuff that's making it smell is embedded in the plastic/rubber and might need to be neutralized with something like a baking soda soak* every few days. Once the smell is there, it never really goes away, in my experience.

Maryn, veteran of much orthodontia, and her kids'

*Buy a box of baking soda (not baking powder). Find a container that your appliance will fit in, like a drinking glass or a cup, and fill it up to about an inch from the top with lukewarm-to-warm water. Spoon in baking soda and stir until no more will dissolve and you can feel a sludge of it starting to form at the bottom. Drop in the appliance and leave it for at least four hours. Rinse before putting it in your mouth. Discard baking soda and wash container before re-use.