Gemara

(Rav Papa): If someone threw a clump of earth at a date tree and knocked down dates, he is liable for Tolesh (a Toldah of reaping).

(Rav Ashi): [He is exempt.] This is not the normal way to detach.

74b (Rav Menasheh): If one chops wood finely for kindling, he is liable for grinding.

(Rav Ashi): If he is particular about their length, he is liable [also] for cutting.

146a (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).

Beitzah 33b (Beraisa): One may roll soft fragrant wood or cut it, to smell it. One may not cut hard wood. If he cut it, he is exempt, but it was forbidden. He may not cut it to clean his teeth. If he cut it, he is Chayav Chatas.

(Rava bar Rav Ada): Rav Yehudah used to cut and give us give big sticks, even though it could be used for handles for axes and chisels.

(Beraisa - R. Eliezer): One may take a chip from in front of him to clean his teeth;

Chachamim permit only from an animal's trough.

Both forbid cutting it.

R. Eliezer says, if he cut it on Shabbos to clean his teeth or open the door, he is Chayav Chatas;

Chachamim forbid only due to Shevus.

There R. Eliezer is Mechayav Chatas, so here he forbids l'Chatchilah. There, Chachamim forbid only due to Shevus, so here they permit.

Rishonim

Rambam (Hilchos Shabbos 11:7): Any cutting of wood or metal is liable for Mechatech. If one takes a chip from in front of him and cuts it to clean his teeth or to open the door, he is liable.

Magid Mishneh: Why does the Rambam obligate? Rashi says that Chachamim say that it is a proper Tikun only if he uses a knife. If the Rambam discusses cutting with a knife, he should have specified.

Rosh (Beitzah 4:13): Rabanan say that if one used a knife to cut a chip to clean his teeth, he is Chayav Chatas. They decreed to forbid by hand due to with a Kli.

Rashi (73b DH Ein): It is not normal to detach via throwing, rather, by hand or with a Kli. Via throwing is k'Le'acher Yad (unskillfully), so he is exempt.

Poskim

Shulchan Aruch (OC 322:5): One may move fragrant wood to smell it or wave it for a Choleh.

Magen Avraham (5): The Gemara permits for a Choleh. This connotes that it is forbidden for a healthy person. Perhaps this is merely the typical case. See 308:23. (If one cut a Lulav before Shabbos, he may use it to chase away flies, for he made it a Kli.)

Gra (4): There are three laws. One may move and cut animal food, even to clean his teeth. One may move and cut fragrant wood to smell; to clean his teeth is exempt, but forbidden. One is exempt also for a chip, but one may not even move a chip.

Mishnah Berurah (15): From the beginning, he did not divert his mind from this wood.

Mishnah Berurah (17): Rolling it between the fingers causes the smell to exude. Also cutting it causes the smell to exude from where he cuts. Many Acharonim permit cutting only by hand, even though it is in order to smell, and he has no intent to make a Kli. This is a decree lest he cut with a Kli to make a toothpick. Even by hand, one may cut hard wood only to smell, but not to clean his teeth.

Shulchan Aruch (ibid.): He may roll it to smell it, whether it is hard or soft.

Beis Yosef (DH u'Mah): Chachamim permit cutting even hard wood to smell it. This is like the Mishnah that permits breaking a barrel to eat the figs, as long as he does not intend to make a Kli. Rav Yehudah used to cut and give big sticks, even though they could be used for ax handles. He cut them for smelling.

Taz (3) and Kaf ha'Chayim (19): The text of the Shulchan Aruch should say "he may cut it, and roll it to smell it", like it says in the Tur and Beis Yosef.

Taz (3): The Yam Shel Shlomo (Beitzah 4:8) permits cutting it only by hand, but not with a knife. It was permitted with a knife only regarding animal food. I wrote (519:5) that we do not hold like this.

Magen Avraham (6): See Sa'if 4 and what I wrote in 511:11. One may increase the smell, but one may not create a smell.

Eshel Avraham (6): It seems that even by hand, it is permitted only to smell, but to clean his teeth it is at least forbidden. Eliyahu Rabah (7) brings that Shirei Keneses ha'Gedolah forbids to cut and give to another. This is difficult. Rav Yehudah broke off big pieces and gave to others! We cannot prove from here whether one may break off a leaf from a Lulav on Yom Tov to bind it with a knot permitted on Yom Tov. Here we discuss fragrant wood.

Kaf ha'Chayim (21): Many permit breaking off to give to another. People break vine sprigs off a cluster to give to others. Ben Ish Chai says that this is the custom, and one who is stringent will be blessed.

Mishnah Berurah (18): Even if they are hard and unsuitable for animal food, one may cut a big or small branch, but only to increase the smell. One may not make a new smell, e.g. to put fragrances in a garment to scent the garment.

Kaf ha'Chayim (22): Breaking myrtle branches to fix them to put with the Lulav is like to clean teeth. It is permitted only if they are soft. One may clean his teeth with fragrant wood without breaking it.

Kaf ha'Chayim (23): One may clean his teeth with a Lulav leaf used to tie the species together even after it was removed, only if as it is moist and could be used to tie in a way permitted on Shabbos, for it is still a Kli.

Mishnah Berurah (18): Here are other laws of cutting on Shabbos. 1) If one cuts hide or anything that one cuts and he is particular about the size, whether wood, metal or even bird feathers, since he is particular about the length and width and intentionally cuts, he is liable. However, if one cuts wastefully, or without intent for the length and width, rather, like Misasek or playing, he is exempt. How great were the Nevi'im and Chachamim, who forbade us to buy and sell on Shabbos, lest one come to a Chiyuv Chatas! E.g. he sells merchandise and cuts it to the needed size. 2) Mechatech does not apply to food, like I proved above (12). 3) In the Bi'ur Halachah I wrote that Mechatech depends on each matter, how it is done. If something is normally cut with a Kli, one is liable only with a Kli. If something is separated by hand, one is liable even by hand. Presumably, in every case it is forbidden [at least] mid'Rabanan. If one uses a knife to cut a chip to clean his teeth, the Rosh obligates for Tikun Kli, and the Rambam obligates for Mechatech. Perhaps the Rambam obligates him twice [also for Tikun Kli].

Bi'ur Halachah (DH Echad): Why did Chachamim permit cutting hard wood to smell? Granted, he does not intend to make a Kli, so Tikun Kli does not apply, but it should be forbidden due to Mechatech hard wood, which is not animal food! The Melachah of Mechatech is not due to making a Kli. The cutting itself is the Melachah. If one chops wood finely for kindling, if he is particular about their length, he is liable [also] for cutting (74b). Perhaps since the Shulchan Aruch discusses cutting by hand, Mechatech does not apply, for Stam Mechatech is with a Kli. The Rosh connotes like this. He exempts one who cuts by hand in order to clean his teeth, for this is k'Leacher Yad. Also Rashi says that cutting without a knife is not a proper Tikun. It is k'Le'acher Yad. If Mechatech applied even by hand, he should be liable for Mechatech! However, Nishmas Adam says a Chidush, that Mechatech applies even by hand. This is difficult. Why did Chachamim permit cutting hard wood? Perhaps Stam cutting is to smell, and he is not particular about the size. He intends only that cutting will increase the scent. However, if he is particular about the size, e.g. his friend requested fragrant wood and he cuts a Shi'ur in order to give to him, indeed he can transgress a Torah Isur.

Bi'ur Halachah (ibid.): The Rambam holds that Stam, one who cuts to clean his teeth is particular about the size, for he obligates due to Mechatech. I.e. he wants it to be smooth, and good for cleaning teeth. The Magid Mishneh holds like the Rosh, that since there is no Chiyuv for Mechatech, this shows that Mechatech is only with a Kli. This is not difficult for Nishmas Adam, for he tried to explain the Rambam with a Chidush without a proof. I say that it depends on each matter. If something is normally cut only with a Kli, one is liable only with a Kli. We find that one is exempt for Gozez (shearing) by hand, since this is abnormal (94b). Something normally separated into two by hand, e.g. merchandise that one tears into two by hand, one is liable for Mechatech by hand, like we find regarding Tolesh (detaching - Rashi 73b). All this is if he is particular about the size. Regarding a Chiyuv mid'Oraisa, I say that each matter is based on what is normal for it. Mid'Rabanan it is forbidden in every case, just like shearing. All forbid it by hand, even though it is abnormal. If so, we must say that the Heter to cut in order to smell is because Stam, he is not particular about the size. Perhaps Mechatech does not apply at all by hand. However, presumably for something normally separated by hand, Mechatech applies by hand. Therefore, in any case one should not be lenient.