Gemara

75b (Rabah bar Rav Huna): If one salted meat, he is liable for tanning.

(Rava): [He is exempt.] Tanning does not apply to food.

(Mishnah): Memachek (scraping hide) and cutting [it into strips is an Av Melachah].

(Rav Acha bar Chanina): If one rubs between pillars, he is liable for Memachek.

(R. Chiya bar Aba): If one smears a bandage on a wound, he is liable for Memachek.

124b: Rava was walking in a Reshus ha'Rabim. His shoes got muddy, so he took a piece of earthenware and cleaned them.

141a (Beraisa): If there is mud on one's shoe, one may scrape it off with the back of a knife. If there is mud on one's garment, one may scrape it with his fingernail.

(R. Avahu): One may scrape a new shoe, but not an old shoe. He may use the back of a knife.

Objection (an elder - R. Chiya's - Beraisa): One may not scrape a new or old shoe.

Rishonim

Rambam (Hilchos Shabbos 22:18): If there is mud on a shoe or sandal, one may shake it in water, but one may not launder it. One may not scrape a new shoe or sandal, but one may anoint it, and clean an old shoe or sandal.

Rosh (20:14): Rav Kahana permits scraping mud on a garment with a fingernail or knife.

Tosfos (75b DH ha'Shaf): Rav Acha forbids scraping hide against a pillar, to smoothen it. R. Chananel and the Yerushalmi say so. Rashi says that he smoothes the ground between pillars. This is wrong.

Poskim

Shulchan Aruch (OC 302:8): One may not scrape a shoe with a knife or his fingernail, whether the shoe is old or new, for this peels the hide, and this is Memachek

Beis Yosef (DH Ein): The Beraisa connotes that one may not scrape a shoe even if there is mud on it. Therefore, the Rif, Rambam and Rosh omitted R. Avahu's Heter. The Tur brings the Beraisa that permits with the back of a knife. The Kolbo says that this is only for dry mud, since scraping it fixes the shoe. One may scrape wet mud, since this is not Tikun. R. Peretz forbids scraping dry mud off a garment. The Heter to scrape with the fingernail is only for wet mud. If so, the Reisha, which permits mud on the shoe with the back of a knife, must be even for wet mud. Even so, it is rejected due to R. Chiya's Beraisa. It forbids Stam, which connotes even wet mud.

Magen Avraham (17): It is forbidden with a knife, but one may scrape it on the wall, like it says in Sa'if 6. Maharil used to clean his shoes on an iron [bar] fixed in front of the Beis ha'Keneses to scrape the soles from mud. I say that it was thick and wide at the end, and not sharp. If it was sharp at the end, this is like a knife that peels hide. In any case, one should be stringent, for even the back of the knife is forbidden. The Bach forbids with a beam.

Gra (DH Oh): The Gemara permits scraping a garment with a fingernail. The Rosh and Tur say that all the more so, one may use the back of a knife. If so, since we forbid the back of a knife for shoes, all the more so a fingernail is forbidden.

Mishnah Berurah (37): This is even if there is mud on it and he intends to remove the mud, for it is a Pesik Reishei [that he will be Memachek].

Mishnah Berurah (38): Even the back of a knife is forbidden. Therefore, the iron [bar] in front of the Beis ha'Keneses to clean soles should be thick and wide, and not sharp. One may clean on the wall.

Kaf ha'Chayim (59): Korban Nesan'el (Shabbos 20:20) says that all permit scraping wet mud off the bottom of the shoe. Yad Efrayim says that only scraping is forbidden. Maharil used to clean gently, lest he come filthy with mud into the Beis ha'Keneses. Normally it is good to be stringent like the Bach, who forbids even with a beam. If one cleans the bottom gently with a thick iron bar, perhaps one may be lenient due to honor of the Beis ha'Keneses, or if he will enter the house of a great person.

Bi'ur Halachah (DH Ein): The Beis Yosef discusses according to R. Peretz. R. Peretz' law is not clear. The Mechaber brought it as "some say..." in Sa'if 7. Permit the other Poskim hold like the Kolbo! The Bach questioned the Kolbo, for Rashi explained that the Beraisa forbids due to Memachek. If so, the same applies to wet mud. How can the Kolbo permit what Rashi and all the Poskim forbade?! This is not a rejection. It seems that the Rambam explains that it is not due to Memachek, for he taught this with laws of laundering, and not with laws of Memachek! A support for the Rambam and Kolbo is the Beraisa, which taught together about mud on shoes and on garments. Even according to Rashi and the other Poskim, who forbid due to Memachek, perhaps this applies only to dry mud, which must be scraped with force, and he will come to peel the hide. This does not apply to wet mud. I later found that the Me'iri forbids due to Memachek, but only if the mud is dry. One may rely on them if there is another reason to permit, e.g. he cleans gently. All permit on a wall.

Kaf ha'Chayim (60): Olas Shabbos says that surely R. Peretz forbids even wet mud. He forbids dry mud [even on garments] due to grinding, so the Isur [on shoes] must be even if it is wet!

Rema: One may remove the mud with a shard that is not Muktzeh.

Kaf ha'Chayim (46,47): All permit this, but only regarding feet. It is forbidden regarding shoes because it is Memachek.

Rema (321:19): One may smooth food on Shabbos. There is no concern for Memachek, since one could eat it without smoothing it. In any case, one who is stringent regarding a food of apples and similar things that are normally smoothed, he will be blessed.

Magen Avraham (29): If one could not eat it without smoothing it, it is forbidden, for mid'Rabanan, tanning applies to food. Smoothing is like Memachek.

Gra (DH u'Mutar): The Gemara says that tanning does not apply to food. The Mordechai (362) derived that one may smooth Tartarish (a food made from apples) without concern for Memachek. In any case there is no proof from here, like Tosfos (75b DH Ein) says [that mid'Rabanan, tanning applies]. In any case it is permitted, for one could eat it without smoothing. We learn from Ze'iri (109a. He permits straining clear wine or clear water through a cloth, and scrambling soft-boiled eggs, since one could drink or eat them without doing so.) One who is stringent will be blessed.

Mishnah Berurah (80): The Melachah of Memachek applies to wax, pitch and similar things that are smeared.

Mishnah Berurah (81): If one could not eat it without smoothing it, mid'Rabanan tanning, and also Memachek, apply to food. Perhaps it is forbidden even for that very meal.

Mishnah Berurah (82): One may spread cooked apples on bread. Also lard and butter are surely permitted.

Bi'ur Halachah (DH b'Ma'achal): Perhaps the Rema said 'food of apples' to teach that it is proper to be stringent only for an apple dish (and the same applies to potatoes), for which it is normal to beautify and smooth the food, and it looks like smearing. One need not be stringent about other foods at all. This is like the Magen Avraham's Heter to smear on bread, but he was lenient for another reason.

Bi'ur Halachah (DH Tavo): Semak brought that mid'Rabanan, Memachek applies to food. He permits smearing cooked apples in a meat pie, since he does so only to put them where there is none. I.e. he does not intend to smear, rather, to fill the empty place. Even though it is a Pesik Reishei, and we forbid a Pesik Reishei even for mid'Rabanan matters, here one could eat it without this, so one should not be stringent at all. It seems that all agree that mid'Oraisa Memachek does not apply to food, i.e. if he smears the food itself. If one seals a hole in a barrel and smears it above, Memachek applies. Eliyahu Rabah says that Chelev and lard are like wax.

Kaf ha'Chayim (136): If apricots dried out very much, people soak and rub them until they dissolve. There is no problem of Memachek, since one could eat it once it soaked. However, if they were inedible dry, one should not soak them on Shabbos.

Shulchan Aruch (323:9): One may rub Kelim with anything, except for silver Kelim with dry, hardened dregs of wine, for it is Memachek the silver, which is soft.

Beis Yosef (DH u'Mutar): A Beraisa (50a) forbids with hardened dregs, for it is a Pesik Reishei that he will be Memachek. It is permitted with niter or sand, like R. Shimon, who permits Davar she'Eino Miskaven. It seems that one may rub other Kelim with anything.

Magen Avraham (15): Maharil says that the Rosh forbids polishing glass with oats. This requires investigation. Perhaps he holds that one may rinse, and forbids only polishing.

Mishnah Berurah (38): It is permitted only if the sand was designated before Shabbos, so it is not Muktzeh. He may not add water on Shabbos, due to kneading, unless there is so little sand that the particles do not stick to each other. Perhaps Maharil permits only rinsing, so the Kelim will be nice, but not to polish. This is difficult.

Mishnah Berurah (40): One may not sharpen a knife on Shabbos, even not on a sharpener, e.g. with wood, for this is like Memachek.

Kaf ha'Chayim (66, citing Taz 326:3): If lard hardened on the Kli, rubbing it will melt the lard. Since he does not intend for this, we do not forbid due to Nolad.