Laundering is one of the acts prohibited on Shabbat based on its connection to bleaching.  Bleaching was required to prepare the wool for the curtains of the Tabernacle and laundering is an activity related to bleaching. It is good news then- those loads of laundry will need to wait until Sunday.

The prohibition however applies only to materials that are somewhat absorbent, meaning your dishes can be cleaned, as can your wood, metal or ceramic surfaces.

Soaking both dirty and clean fabrics is also prohibited as part of the laundering process.  The prohibition does not include soaking non-absorbent surfaces or to soaking non-absorbent fabric in dirty water. The acceptance of soaking in a dirty liquid makes room for the allowance of activities such as drying hands on a towel and wiping dishes, as the towel becomes dirty rather than clean.

As wringing is part of the laundering process, it too is forbidden.  Under Biblical law the prohibition included only absorbent fibers like wool, cotton in linen. The Rabbinic Sages extended the prohibition to include less absorbent fibers that may trap water, like steel wool and hair on the head and beards.

It is forbidden to remove a stain from fabric with any type of cleansing agent, even plain water. The removal of a stain by rubbing or scraping is also prohibited though in cases of absolute necessity, a gentile may be asked to rub out a stain.

Should one’s clothing become soiled by wet mud or dirt, it may be scraped before it dries so long as a stain will remain on the fabric.   

For spills, a garment, such as an apron, may be used to wipe up small spills so long as they are not large enough to require that the garment be wrung out. For larger spill, a towel or rag needs to be used instead as these are fabric items that one would not need to wring out and would be able to leave soaking or saturated.

Should your clothing become wet in the rain, you can hang them to dry over a chair or in any other atypical place. Use of one’s laundry room or dry rack for these items that have become inadvertently wet is forbidden because it may give the appearance that the laws are being violated.

Information obtained from The Shabbos Home, Rabbi Simcha Bunim Cohen