Gemara

(Mishnah): One may break a barrel to eat the dry figs inside, as long as he does not intend to make a Kli (i.e. a nice opening);

If a barrel was punctured, one may not put wax on it.

146b: Rav decrees to forbid oil, lest one put wax. Shmuel does not decree.

147b (Beraisa): One may not scratch himself with a scraper on Shabbos;

R. Shimon ben Gamliel says, if his feet were dirty, he may scrape them normally.

Rav Shmuel bar Yehudah's mother made a silver scraper for him.

Rishonim

Rambam (Hilchos Shabbos 11:6): One who smears a bandage of any size, or wax or pitch or similar things that smear until they are smooth, is liable for erasing.

Rambam (23:11): One may not seal a hole with wax or similar things, lest he smear. We decree to forbid even with lard, due to wax.

Poskim

Shulchan Aruch (OC 314:11): One may not put wax or thick oil in a hole in a barrel to seal it, for he smears. One may put other things, for which there is no concern for smearing.

Mishnah Berurah (45): Even though smearing does not apply to oil, we decree to forbid, due to wax. This applies only to thick oil, for smearing applies to it somewhat. One could come to confuse them. Chelev and lard are like wax.

Shulchan Aruch (OC 326:10): One may wash his hands with bran.

Rema: Kneading k'Le'acher Yad (unskillfully) is permitted. One may not wash with soap or other Chelev that dissolves on his hands, for it is Nolad.

Shulchan Aruch (327:3): One may not scratch himself on Shabbos with a Kli special for this, unless his hands or feet were dirty.

Magen Avraham (1): If one had a special scraper for Shabbos, it is permitted. This led to the custom of single girls to have a brush from pig's hair special for Shabbos.

Mishnah Berurah (10): Since they have a special Kli for Shabbos, it is not Uvda d'Chol.

Mishnah Berurah (9): One may scratch with his hands on the flesh, but he must be careful not to make a wound.

Igros Moshe (OC 1:112 DH uv'Davar): Surely one may not brush his teeth with toothpaste, for this is Memachek (smoothing). Without toothpaste, with a brush special for this, it is permitted, like bathing one limb. However, he may rinse only in cold water. It is good not to wet the brush with water before cleaning [his teeth], lest he squeeze hair (i.e. water from the bristles).

Minchas Yitzchak (3:48): The questioner forbids polishing and brushing teeth on Shabbos due to squeezing, uprooting or breaking bristles (like it says in Minchas Shabbos 80:117 regarding a wig), and smearing the toothpaste. I add the Isur to scratch himself on Shabbos with a Kli made for this, due to Uvda d'Chol. Even if the toothbrush was special for Shabbos, regarding a hard toothbrush we must be concerned for wounding. This is common when the brush touches the gums above and between the teeth. See Mishnah Berurah 327:9. (One who scratches with his hands must be careful not to make a wound.) All the more so this applies to a toothbrush. It is almost impossible to be careful about this. The only Heter is to brush without water or toothpaste, in a way that there is no concern for uprooting bristles or wounding, and with a Kli special for Shabbos.

Mateh Levi (2:21, cited in Tzitz Eliezer 7:30): We are concerned for smearing only in cases like a bandage on a wound, which is hard, such as a band-aid, but not regarding an ointment, which is soft. It is like anointing. Washing hands with soap is like smearing. Even those who forbid, forbid only because he causes the soap to dissolve, and this is Nolad. An ointment is not Nolad, dissolved or smeared. It is like Chelev, which is forbidden only due to Nolad, and here it is not Nolad.

Tzitz Eliezer: There are many kinds of ointments. Some are thick, and smearing applies. Even thin ones can thicken over time. We cannot allow women and Amei ha'Aretz to decide an issue that can lead to a Torah Isur. We find that Rav decreed not to put thick oil on a punctured barrel, even though it need not be smeared, lest one put wax, which must be smeared. The Rambam and Shulchan Aruch rule like this. All the more so there is at least an Isur mid'Rabanan of smearing ointment. However, the Beis Yosef (328 DH Isplanis) says that smearing does not apply to a very soft ointment. This requires investigation. The Mishnah Berurah says that Chelev and lard are like wax.

Tzitz Eliezer (5): Ginas Veradim (OC 3:14) says that the Rema forbids soap due to Nolad, but not due to smearing, for it is Einah Tzerichah l'Gufah and Davar she'Eino Miskaven. One cannot learn from there to here, for here he intends to smear. The Mishnah Berurah (326:30) says that all forbid our soap, which is soft, due to Memachek. The same applies to toothpaste.

Tzitz Eliezer (7): The Magen Avraham (316:24) said that one may rub spit into the ground because smearing does not apply when one wants it to be absorbed. This shows that he was concerned for spit, even though there is nothing softer! The Eliyahu Rabah agreed; Kaf ha'Chayim (105) questioned this from the Beis Yosef, who says that smearing does not apply to a very soft ointment. One is liable for any amount of smearing (Rambam).

Ketzos ha'Shulchan (7:138, Badei ha'Shulchan 130) permits brushing teeth when one uses his hands to put the toothpaste on the teeth. It is not Refu'ah, and there is no smearing. There is no Nolad, for the toothpaste is soft and remains soft. It is Uvda d'Chol to put the toothpaste on the brush.

Tzitz Eliezer: I disagree, like I explained. Also, when he cleans the brush afterwards, he squeezes the bristles by hand. Sometimes brushing causes the gums to bleed. Ohel Moshe (2:98) forbids due to smearing, and possibly due to Refu'ah. Toothpaste is made for medical reasons to remove plaque. Nolad applies, for it contains soap. Perhaps it is a Pesik Reishei that he will bleed. I hold that it is not Refu'ah, for all healthy people do so. It is not a Pesik Reishei that he will bleed. Perhaps it is Nolad, for the toothpaste dissolves in the mouth when one rinses with water.

Yabi'a Omer (4 OC 29:16): Using toothpaste is not Refu'ah. It prevents damage due to decay of food left between teeth.

Yabi'a Omer (4 OC 30): Ketzos ha'Choshen forbids using a brush due to Uvda d'Chol. The Rema (337:2) forbids using a clothes brush lest the bristles break. He was not concerned for Uvda d'Chol. Maharikash says that no one is concerned for this, for breaking bristles is not a Melachah, and he does not intend for this. The Taz questioned the Rema, for even if he breaks bristles, this is Mekalkel (destructive) and therefore exempt. Ma'amar Mordechai questioned the Taz, for he is exempt, but it is forbidden! Rashi (146a DH Shover) permits breaking a barrel, for there is no Isur be Mekalkel. The Rashba and Ran forbid Mekalkel l'Chatchilah. However, we permit regarding a barrel for the need of Shabbos. Even though the Rema forbids a brush lest the bristles break, this is not a Torah Isur. Tosfos (103a DH Lo) permits l'Chatchilah a Pesik Reishei d'Lo Nicha Lei of an Isur mid'Rabanan. Many Acharonim rule like this. Therefore, it is permitted in any case for the needs of Shabbos. Further, the Beis Yosef (Sof Siman 317) brings from Shibolei ha'Leket 87 that there is no Isur Shabbos at all to break off something detached, unless he cuts it to size or makes a Kli. Also Maharam mi'Rotenburg says so. He equates this to seals on Kelim. One may untie or cut them, unless it is Tikun Kli; therefore one may cut strings [that hold stuffing] inside roasted fowl. The Magen Avraham (314:14) and Shulchan Aruch ha'Rav (340:17) bring this. Tosefes Shabbos permits only for needs of Shabbos. This answers the question of Eshel Avraham (317:20), why one may not tear without intent to sew, which is Mekalkel on something detached.

Yabi'a Omer: Therefore, one may use toothpaste on a toothbrush. It is like breaking something detached, which is permitted. Do not say that the Rema forbids a clothes brush because he equates breaking bristles to cutting something attached. Even the Taz (336:10), who forbids pulling off skin because it is as if it is attached, says so only because man is equated to land. Also, Tal Oros and the Bi'ur Halachah (340:2) brought Rishonim unlike the Taz' reasoning. Also, it is a Pesik Reishei d'Lo Nicha Lei, and he breaks the bristles k'Le'acher Yad. The Ramban and Rivash (387) say that k'Le'acher Yad is more lenient than other Shevusim. Normally we permit Shevus of Shevus only for a Mitzvah. Here is more lenient, for it is k'Le'acher Yad. The Sha'ar ha'Tziyun (337:2) says that surely we are lenient about a Pesik Reishei d'Lo Nicha Lei of a Shevus d'Shevus. Also the Chazon Ish (Mo'ed 50:5) is lenient. Also, perhaps it is not a Pesik Reishei. Often, no bristles break!

Yabi'a Omer: The Bach (337) and Chazon Ish (ibid.) were concerned for Uvda d'Chol. The Beis Yosef and Maharikash were not, for wherever we do not explicitly find that Chazal were concerned, we are not concerned. Alternatively, a need of Shabbos is like a Mitzvah, which overrides Uvda d'Chol (Rema 333:1, Mishbetzos Zahav 333:1, Bi'ur Halachah 333:1 DH v'Chol, 338:1 DH Ho'il). One who regularly uses toothpaste during the week has pain if he refrains on Shabbos. Igros Moshe is concerned only for smearing. I say that also this is not a concern.

Yabi'a Omer (19): However, one must wet the brush, and it is a Pesik Reishei that water will be squeezed. Many hold that mid'Rabanan, squeezing applies to hair. This is one Safek, and perhaps the Halachah follows those who permit a Pesik Reishei of a mid'Rabanan law. We are lenient due to the Sefek-Sefeka, especially since this is k'Le'acher Yad, and it is not the normal way to squeeze. Ketzos ha'Shulchan (6:133, 16) forbids wetting a brush to wash his body due to a Pesik Reishei of squeezing, and perhaps soaking the brush is laundering. We hold that "soaking is laundering" only for something dirty. Even those who forbid squeezing hair forbid only mid'Rabanan, so we do not decree lest one squeeze (Avnei Nezer 1:169). Also, here he is not concerned for the water squeezed out. One may put even much water on a garment in such a case (Mishnah Berurah 320:47 and other Acharonim there).

Yabi'a Omer (20): Even those who do not use soap on Shabbos may use toothpaste. However, if one always bleeds, it is forbidden. One should designate a special brush for Shabbos, and not wash it afterwards. If one can refrain from brushing, and will not be pained, it is good to be stringent.