It is possible to clean a coffee-stained stainless steel mug with simple ingredients you can find in your kitchen - baking soda and vinegar. Add baking soda to the stainless steel mug. Afterward, pour the white vinegar in it and allow it to form bubbles. When the bubbles finally settle, cover the mug with its lid and shake. Toss out the mixture and rinse your mug. The most important thing to remember with this coffee stain removal method is that you need to let the bubbling stop before covering the mug. This will prevent the chance of explosion due to the reaction of baking soda and vinegar.