What a great question! Both locations are places that tend to be warm, dark, and damp, with organic molecules available for food. Consequently, I would expect to find that both have lots of fungi present. Fungi are decomposers, meaning that they break down either living or dead organic matter to get energy. In the outdoor scenario, the rotting log itself is the food source, while inside a running shoe, the food source may be more variable - it may or may not include components of the shoe itself (this depends on whether the shoe is made out of natural or synthetic materials), but there will also be dead skin cells and various proteins and other organic components of sweat available.
The other life form I would expect to see plenty of would be bacteria, which also are important decomposers and would be doing the same things as the fungi.