One of the 39 Melahot prohibited on Shabbat is "Melaben," which means whitening. This generally refers to cleaning raw wool or flax from their impurities. One of the Toldot (sub-categories) of Melaben (Whitening) is "Koves" (Laundering). This includes soaking fabric in water or wringing out an already wet fabric. The "endgame" of these actions is the same as Melaben-the garment becomes cleaned. This is the way the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) explains these categories in his introduction to these Halachot in Siman 302.
The Shulhan Aruch holds that the restrictions of cleaning leather are more lenient than with regard to fabric. Accordingly, Maran rules that if one’s leather shoes are dirty, he is allowed to run water over them, whereas with cloth shoes, one is not allowed to, because of the principle "Shriyato Zehu Kibuso"-applying water to cloth is tantamount to laundering. Scrubbing the shoe with water is prohibited even on leather shoes.
Another example of a leniency with leather is dusting off a shoe. Maran permits taking a dry towel to remove the dust, as long as one does not rub hard and shine the shoe, whereas doing so to a cloth garment is problematic.
One should note that sometimes a leather shoe has segments containing cloth. In such cases, it is problematic to apply water to the shoe, as the water may soak the cloth part, as well. External stitching of cloth fibers also constitutes a problem, whereas internal stitching does not.
There is no issue of laundering whatsoever with regard to plastic or rubber shoes, such as Crocs or galoshes. Therefore, it is permissible to not only apply water, but to scrub the shoe as well.
It goes without saying that one may not polish any type of shoe on Shabbat.
Hacham Ovadia and the Shemirat Shabbat K’hilchata permit soaking contact lenses overnight in their solution. This type of rinsing is not defined as cleaning, but is done merely to keep it moist so that it doesn’t dry up.
SUMMARY It is permitted to apply water or dust off a leather shoe on Shabbat. However, it is prohibited to polish and shine a leather shoe or scrub it with water. If the leather shoe has cloth components, the shoe is regarded as a garment and these leniencies do not apply. No restrictions apply to cleaning a rubber or plastic shoe.
It is permitted to soak contact lenses in their solution on Shabbat.