Gemara

(Mishnah): We may not clear out Luf or mustard;

R. Shimon permits Luf, for ravens eat it.

128a (Beraisa): One may move squill because deer eat it, and mustard because doves eat it;

R. Shimon ben Gamliel says, one may even move broken glass, because ostriches eat it.

Objection (R. Noson): If so, you should permit moving also bundles of twigs, because elephants eat them!

Answer (R. Shimon ben Gamliel): Ostriches are common. Elephants are rare.

(Ameimar): R. Shimon ben Gamliel permits broken glass only if he has an ostrich.

Objection (Rav Ashi): If so, why did R. Noson ask about bundles of twigs? If he has an elephant, indeed he may move them!

Rather, R. Shimon ben Gamliel permits broken glass to everyone, for it is proper to have an ostrich. R. Noson asks that the same should apply to bundles of twigs, for it is proper to have an elephant!

(Abaye): The following Tana'im hold that all of Bnei Yisrael are princes: R. Shimon ben Gamliel, R. Shimon, R. Yishmael and R. Akiva.

(Beraisa): One may move bones, for dogs eat them;

156b (Mishnah): We may cut a Nevelah in front of dogs [on Shabbos];

R. Yehudah says, if it was not dead when Shabbos began, we may not cut it, since it was not prepared.

Rishonim

Rif and Rosh (50b and 18:2): Whenever the Gemara says 'the following Tana'im hold...', this is a Shitah, and the Halachah does not follow it. One may not move mustard, like the Stam Mishnah.

Ran (DH u'Maskana): One who has ostriches and elephants may move broken glass and bundles of twigs. R. Shimon ben Gamliel is more lenient (this is the Bach's text), and permits broken glass for everyone, since princes normally raise them. The Halachah does not follow him. One may not move a food proper for Chayos or birds unless he has them in his Reshus, or if people normally raise them. The Beraisa permits mustard, because doves eat it. This is unlike our Mishnah. Even so, the Rif and Rambam permit. They hold that the Tana'im do not argue. Each discusses his locale. Ravens were commonly raised in the locale of the Tana of the Beraisa, but not in the locale of the Tana of the Mishnah. All agree that one may move only food for animals commonly found among people.

Rif (59b): One may move bones only if they are proper for dogs and animals. If not, R. Shimon forbids them.

Ran (DH Gemara): One may move bones proper for dogs, even if they separated [from the meat] on Shabbos. If they came off before Shabbos, even R. Shimon agrees [that one may not move them].

Question (Beis Yosef OC 308 DH v'Chasav Od): Why is this different than a Nevelah that died before Shabbos, which even R. Yehudah permits (156b)? Also 'even if they separated on Shabbos' connotes that all the more so if they came off before Shabbos, they are permitted. Reasoning says just the opposite! Perhaps the text should say 'R. Yehudah' in place of R. Shimon. I.e. even R. Yehudah permits a Nevelah that died before Shabbos, and likewise bones that came off before Shabbos, since they were prepared for animals before Shabbos.

Note: Bach corrects the text like the Beis Yosef said.

Darchei Moshe (308:10): What is the Beis Yosef's question? One may not move bones that came off the food before Shabbos unless he has dogs, just like one may move Luf only if he has ravens. If a Nevelah died before Shabbos, R. Yehudah permits only cutting it up in front of dogs. (Other than this, one may not move it!) The Ran is correct. One may remove from the table only bones that separated on Shabbos, since they are still proper for dogs, even if he has no dogs. One may not move other bones. The Beis Yosef holds like he brought from the Ran, that the Rif and Rambam hold that if ravens are commonly found, anyone may move Luf. If so, the same applies to bones proper for dogs. However, many disagree.

Darchei Moshe: The Beis Yosef asked a second question because he understood "even if they separated on Shabbos, and all the more so if they separated before Shabbos." Really the Ran means "even if they separated on Shabbos, and all the more so if there is still meat on the bones."

Rambam (Hilchos Shabbos 26:16): One may move anything proper to feed to animals and birds. One may move mustard, for it is food for doves, and bones because they are food for dogs.

Magid Mishneh: Why do the Rif and Rambam permit mustard? The Stam Mishnah forbids, and no one argues! The Tana forbids because most mustard is [destined] for people, and now it is not proper for people! Even though it is food for doves, one may not move it. Something proper for people, people do not give it to doves! Also the Rashba says so. He rules like the first Tana. This requires investigation.

Rambam (17): One may not move broken glass, bundles of twigs or Luf, even though they are food for ostriches, elephants and ravens, respectively, since these are not commonly found among people.

Rosh (18:2): The Halachah follows the Stam Mishnah, and not R. Shimon ben Gamliel.

Tosfos (128a DH Metaltelin): According to R. Yehudah, who forbids Muktzeh, we discuss when there was no meat on the bones before Shabbos.

Poskim

Shulchan Aruch (OC 308:29): One may move anything proper to feed to commonly found Chayos and birds. If it is proper only for rare Chayos and birds, if he has such a Chayah or bird, he may move food proper for it. If not, he may not.

Gra (DH Kol): The Beraisa permits one who has an elephant to move bundles of twigs. Even though the Halachah does not follow R. Shimon ben Gamliel, because ostriches are common only among princes, [one may move food proper for animals he has].

Mishnah Berurah (119): It must be commonly found among most people. It is not enough that rich people raise them.

Rema: Based on this, one may move bones that came off meat before Shabbos if they are proper for dogs, since dogs are commonly found (Beis Yosef, unlike the Ran).

Magen Avraham (54): The Darchei Moshe holds that if they separated before Shabbos, the Ran permits only one who has dogs. Therefore, here he permits what separated on Shabbos. Surely, if dogs are not commonly found, it is forbidden in every case.

Gra (DH u'Lefi): The Beis Yosef is correct. One must fix the text of the Ran. Also Tosfos (128a) holds like this.

Mishnah Berurah (120): If such Chayos and birds are commonly found, even one who does not have them may move food proper for them.

Mishnah Berurah (121): All the more so, what separated on Shabbos is permitted, for at the beginning of Shabbos it was [part of something] destined for people to eat. If dogs are not commonly found in the area, one may not move the bones.

Bi'ur Halachah (DH Beis): The Rema, like he wrote in Darchei Moshe, holds that this is unlike the Ran. The Bach and Gra hold that the Ran agrees to this. Darchei Moshe said that many argue with the Rif and Rambam, who hold that if Chayos and birds are commonly found, even one who does not have them may move things that they eat. Ma'amar Mordechai did not find anyone who disagrees. The Shulchan Aruch rules like the Rif and Rambam, and the Rema did not mention an opposing opinion.

Kaf ha'Chayim (194): If one spent Shabbos in a field, hotel [or boat] where there is no dog, one may not move bones, whether they separated before Shabbos or on Shabbos. One may not move very hard bones that are not proper even for dogs. They are like wood and rocks.

Ohr l'Tziyon (2:26:7): If animals are common within a distance that is normal to walk, this is called commonly found, but not if they are at the other end of a big city and it is not normal to walk to there.