The Shulhan Aruch (302:6) discusses the prohibition of scraping one’s shoes on the ground to remove the dirt or mud from them. By doing so, one may come to smooth the ground and fill holes, which constitutes a violation of Shabbat. However, it is permitted on a paved sidewalk.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) is strict regarding rubbing shoes against a wall or pole, as it appears that one is building. Nevertheless, Rabbi Shlomo Zalman Auerbach (Israel, 1910-1995) writes that nowadays there is no concern. No one will be under the impression that he is building; on the contrary, it looks like he is ruining the wall.
The Mishna Berura discusses the prohibition of rubbing the leather sole of a shoe against something sharp, such as a metal grate, in order to clean it. Doing so is problematic, because it constitutes the Melacha of "Memahek"-smoothing the leather. Hacham Bension rules that today, it is not a problem, since the process of refining leather today is done in a different fashion. There is also no concern regarding the Melacha of Tohen-grinding the dry mud into particles, since it is being done with a Shinui (unusual method).
According to the Shulhan Aruch, it would also be permitted to pour water over a leather shoe. The principle that soaking constitutes laundering does not apply to leather. For example, if someone wants to step into a puddle of water to clean his shoes, he may do so. However, it would be prohibited to scrub the shoe.
It is prohibited to scrape a muddy shoe on the ground, but it is permitted to do so on a paved surface, wall or metal grate. It is also permissible to apply water to a shoe to clean it, as long as one does not scrub it.