The Gemara in Masechet Shabbat, Daf 141, discuses how to remove a garment stain on Shabbat without making any transgressions. The Gemara advises that a person should place his fingers underneath the garment and rub the stain from the inside. This should cause the stain to come apart. While the Gemara says to rub from inside the garment, it is clear that rubbing the garment from the outside is strictly forbidden. Rubbing from the outside is similar to whitening or bleaching, which is the forbidden Shabbat task of Milaben. Furthermore, it is strictly forbidden on Shabbat to use any liquid to remove a garment stain. So rubbing from the inside is permissible, but rubbing from the outside is prohibited, and using liquid agents is prohibited.
The Gemara does allow us to scrape off a stain with a fingernail. Maran in Shulchan Aruch writes this in siman 302. However, he does bring down an opinion in the name of Rabenu Peretz who says the Gemara was not referring to a dry stain but rather a moist stain. According to Rabenu Peretz, rubbing out or scraping off a dry stain in any fashion is a transgression of one of the 39 forbidden tasks on Shabbat known as Tochen (grinding.) So Rabenu Peretz adds a very important stipulation to the Halacha, and says that we are only permissible to work on stains that are still moist and not completely dry.
So, do we abide by the ruling of Rabenu Peretz? Rav Chida says we do not. Rav Chida explains that Maran wrote of this opinion as a secondary opinion (‘Yesh Omrim’). Maran in his main opinion does not differentiate between moist and dry, and therefore we do allow the rubbing of dry stains from inside a garment, or the using of a nail to scrape off a stain.
A good and common example of this is Tehina that falls onto a garment on Shabbat. That Tehina is moist when it falls on the clothing, but it does shortly thereafter harden. So according to the Halacha, even if it is dry, it would be permissible to rub it off from under the garment, or to scrape it off with a finger. Hacham Ovadia Yosef in Halichot Olam prefers the use of a Gentile in this case if possible. Furthermore, he is only lenient if it is a stain that is visible to people and that he doesn’t have another garment. But again, it would be forbidden to use a liquid to remove the stain, and it would be forbidden to rub it from the outside of the garment.
See Mehunat Ahava, Helek 2, Perek 12, Halacha 18.
