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Get a toothbrush specifically for the purpose. Put a small amount of soap on the mouth guard. Use lukewarm water on the toothbrush to create suds. Rinse under warm water. Put into your mouth or into a case.
You can even buy a special cleaning brush, but a toothbrush will work just as well. Don’t use the toothbrush you use for brushing your teeth. Try to find a hard bristled toothbrush. Dishwashing soap is a good choice, but you can also use hand soap. Anti-bacterial soap also works. Gently brush the mouth guard or retainer. Pay special attention to the pockets of dirt and plaque. Make sure all of the soap comes off. Feel the mouth guard to make sure none of the soap remains. Rinse it really well to avoid any unpleasant irritation of the oral mucosa from leftover soap. If your mouth guard or retainer is not in use, it should be in a case. This protects it from getting damaged and away from pets (who often like to chew on it.)