6. One may clean one's hands by rubbing them with crushed grain ("Mursan"), even though one's hands are wet, as long as one doesn't pour the water directly onto the crushed grain (1).
However, it is forbidden, on Shabbos, to rub one's [wet] hands with salt, and certainly, it is forbidden to use soap, because both substances liquify (2).
(1) As we've seen, one of the Avos Melachos (prototype activity prohibited on Shabbos) is kneading ("leesha"), that is, mixing flour and water to make dough. Performing a melacha in an unusual way, is not considered a biblical violation, but it is rabbinically prohibited, except in rare circumstances. However, when using crushed grain to clean one's hands, not only is the "kneading" (mixing the grain and the water on one's hands) being performed in an unusual way, but also, its not one's intent to perform the act of kneading, and, according to many authorities, a biblical act of kneading cannot even be achieved using crushed grain (See Aruch Hashulchan 326:11)
(2) In many circumstances, it is rabbinically prohibited to actively turn a solid into a liquid on Shabbos, because it is as if one has created a new entity; this prohibition is called "Nolad." Using a solid bar of soap involves the additional problem of performing the melacha called "smoothing" ("Memareach"), which entails rubbing or spreading a substance to give it a smooth surface. According to most authorities, it is permissible to use liquid soap on Shabbos, because its fluid consistency exempts it from the prohibition of "smoothing". However, there is an opinion that since liquid soap has some density, it is subject to this prohibition, and thus, to heed this view, one should mix the soap with water (preferably before Shabbos) so that it is extremely fluid ("The Shabbos Kitchen" by Rav Simcha Bunim Cohen, pg 179).