1) To clean them. 2) One of the prohibited activities on Shabbat is kneading, that is, mixing flour and water to make dough. Performing work 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, one doesn't intend to knead, and, according to many authorities, a biblical act of kneading cannot even be achieved using crushed grain (See Aruch Hashulchan 326:11) 3) In many circumstances, it is rabbinically prohibited to actively turn a solid into a liquid on Shabbat, because it is as if one has created a new entity; this prohibition is called ''Nolad'' - new-born. Using a solid bar of soap involves the additional problem of performing the activity of ''smoothing'', which entails rubbing or spreading a substance to give it a smooth surface. According to most authorities, it is allowed to use liquid soap on Shabbat, because its fluid consistency exempts it from the prohibition of ''smoothing''.

1) The Sages prohibited doing anything for healing purposes on Shabbat, except where absolutely essential, because they were concerned that one may come to grind substances for use as medicine; ''grinding'' being one of the 39 prohibited Shabbat activities (see Chapter 80, Law 15). 2) If one stays, it is obvious that one is bathing for medical purposes, otherwise one would not remain in disgusting water for more than a short time. 3) People could assume that the person was unable to find cleaner waters in which to cool off. Remaining in this type of water for a short time is permissible even if one's intent is for healing purposes, as long as it is not obvious to onlookers that this is the intent. 4) Obviously, in the case of hot springs, onlookers can't assume that the person is cooling off, and thus, even immersing for a short time is prohibited. However, according to the Mishna Berurah 328:137, immersing in hot springs for a short time is permissible. Also, the prohibition only applies to someone who is bathing in these springs to alleviate slight discomfort, however, when one is actually ill, it is permissible to bathe in them.

מים שאין הדרך לרחוץ בהם אלא לרפואה כגון שהם רעים או מאוסים אסור לרחוץ בהם בשבת כיון שניכר שהוא מכוין לרפואה במה דברים אמורים כששוהה בהם אבל אם אינו שוהה בהם מותר שאינו נראה אלא כמיקר ובחמי טבריא וכדומה אם אין הדרך לרחוץ בהם אלא לרפואה אסור לרחוץ בהם בשבת לרפואה אפילו אינו שוהה בהם

1) Exodus 23:12. 2) Though the verse specifies a donkey and an ox, it doesn't exclude other types of animals, but is just giving the most common case. 3) This halacha only applies to one's own animal, not an animal owned by a gentile. 4) Even if another person placed the burden on one's animal, one is obliged to remove it.