Keeping your favorite running shoes clean - especially a pair that you've had your eye on for a long time is important. Cleaning your shoes doesn't have to be a loathsome process, especially if you focus on keeping the shoes clean in the first place. Now, we know that your training plan might take you out on some muddy roads and dirty trails. And sometimes life just happens and the sky is clear when you leave the house on your run but a storm rolls in, dumping rain on you and posing dirty conditions for your shoes to endure. You should not be afraid to get out there and get dirty! Instead rinse them with clean, warm water and use a soft cloth or even a toothbrush to scrub away stains. If you tend to run on trails, then keep a bottle of clean water and a rag or extra toothbrush handy in your car so that you can do it right when you get done. When mud, dirt, and debris dry, it makes it harder to get it off.
If your shoe stains are REALLY bad, and if the area of the shoe is rubber or plastic and white, a little bleach can work wonders. Just mix a bit into the water you are using and scrub, making sure to wear some type of gloves to protect your hands from the abrasiveness of the bleach. A milder cleaning solution (i.e. laundry detergent solution) will also do the trick and help you clean your shoes.
If you have gum or sticky stuff stuck to the bottoms of your soles, then work to dig it out with a knife, scissors, or sharp ended tool. Just scrape away at the gum until it has completely been removed. If there are still gum remnants left behind, here is a trick: stick your shoes in a freezer or rub ice over the gummy residue. The goal is to freeze it so that it becomes hard enough to chip away at easier.