Gemara

57a (Mishnah): A woman may not go out with a needle without a hole. If she went out with it, she is exempt.

60a - Question: Why does a woman wear it [even during the week]?

Answer #1 (Rav Yosef): It keeps her hair inside the net.

Objection (Abaye): If so, it should be permitted like Biris (a garter belt to hold up socks)!

Answer #2 (Rav Ada Narsha'ah): She uses it to part her hair.

Question: What does she use it for on Shabbos?

Answer (Rava): It has a gold plate at the other end. On a weekday she parts her hair with the needle. On Shabbos she wears the plate on her forehead.

63a (Mishnah): A Biris is Tahor...

(Rav Yehudah): This is a bracelet.

Question (Rav Yosef - Mishnah): A Biris is Tahor, and one may go out with it;

A bracelet is Tamei! (It is listed among Kelim taken from the war with Midyan, which needed to be immersed!)

Answer: Rav Yehudah means that Biris corresponds to a bracelet (but it is worn on the leg to hold up socks).

(Ravin): Biris is a single [leg bracelet]. Kevalim are a pair.

(Rav Huna): Both of these are pairs. One connects them by a chain to make them Kevalim;

(Rabah bar bar Chanah): There was a family in which the women took large steps, and their Besulim fell out. They made Kevalim for them to limit the size of their steps.

R. Yirmeyah approved of this answer, for also R. Yochanan said so.

Rishonim

Rambam (Hilchos Shabbos 19:5): If a woman went out with a Tachshit (ornament) that it is normal to remove it to show it off, she is exempt. Anything that is a Tachshit and does not fall, and it is not normal to show it off, she may go out with it. Therefore, she may go out with a bracelet on the arm or leg if it clings to the skin and does not fall off. The same applies to everything similar.

Magid Mishneh (DH Lefichach): Rashi explains that Biris supports her socks. She may go out with it, for it is needed for her clothing. We are not concerned lest she remove it to show it off, for she would not expose her leg. This implies that she may not go out with a bracelet, lest she remove it to show it off. The Rambam holds that Rav Yosef comes to explain the difference between them only regarding Tum'ah, but not regarding Shabbos. Even a bracelet is permitted. Biris is permitted, just like a bracelet. They are different [only] regarding Tum'ah.

Rosh (6:7): If the needle is to keep her hair inside the net, it is permitted like Biris. If she would remove it, her legs would not be exposed immediately, only after her socks fall. Rather, it is to part hair. This is a load! She cannot go out with everything she needs, since it is not a garment or Tachshit. Rava answered that the needle has a gold plate at the other end.

Mordechai (353): Biris is on the leg. It supports socks, like a bracelet on the arm. Ravin permits to go out with it on Shabbos. Since it is only on one leg, she will not remove it to show it off, for they will laugh at her that she does not have one on the other leg. Kevalim is when she has on both legs. She would remove it to show it off. Rav Huna said that both of these are on both legs, and she may go out with them, for they are not a Tachshit. This is the Halachah, for R. Yirmeyah and R. Yochanan said so. If so, one may tie straps around her socks lest they fall down, even if they are not tied to them. If they are tied, why is it forbidden when there are two more than when there is one?! They are called Kevalim when a chain connects them. The chain is called a Tachshit. Even though it is made to prevent big steps, it is forbidden, for it is not firmly attached. She could remove it to show it off.

Poskim

Shulchan Aruch (OC 303:9): A woman may go out with a needle without a hole only if it supports the hair net.

Bach (5): This implies that it is even if it is not a Tachshit. This is wrong. It is permitted only if it has a gold plate at one end. Tosfos and the Rosh say so, unlike the Beis Yosef.

Defense (Magen Avraham 8): Tosfos and the Rosh said so about Perush Rashi, regarding a needle to part her hair. If it holds her hair in the net, it is permitted! Rather, it has a gold plate, and it is forbidden lest she remove it to show it off. Even if it also holds her hair in place, it is forbidden lest she remove it to show it off. Even though it is for Tzeni'us we decree, unlike the Beis Yosef.

Shulchan Aruch (15): A woman may go out with socks tied with a string around her thighs, even if it is not tied to them. We are not concerned lest they fall down.

Beis Yosef (DH v'Chasav): Mordechai, Semag, Semak, Sefer ha'Terumah, Hagahos Maimoniyos and R. Yerucham permit.

Magen Avraham (12): Tosfos holds that they [permitted because they] are not a Tachshit. See what I wrote in Sa'if 9. Rashi holds that they are a Tachshit, but she would not remove it to show it off, since it is for Tzeni'us. Therefore, on the arm it is forbidden.

Bi'ur Halachah (DH v'Yotz'ah): The Magen Avraham makes this law dependent on the opinions of Tosfos and Rashi that the Poskim mention in Sa'if 9. The first opinion in the Shulchan Aruch, which is primary, is like Tosfos. The Beis Yosef connotes that he holds like Rashi! However, the Magen Avraham is astounding. He says that the Shulchan Aruch is like Tosfos, who says that Biris is not a Tachshit, and they hold that the same applies to a bracelet. They are totally different! Biris is a ring of Kelim. It serves the socks. Therefore, it is not a Tachshit, and it is Tahor, and she may go out with it on Shabbos. A bracelet is on the arm. It is Tamei, like the verse says. We must say that it is a Tachshit. Why may she go out with it on Shabbos? The Shulchan Aruch cites the Rambam. He clearly connotes that it is a Tachshit! The Pri Megadim says that the Rambam permits a bracelet and Biris because it is not normal to show them off. He is correct. I say that the bracelet is on her arm, under her sleeve. ('It clings' connotes a little like this.) She would not remove it lest she expose her arm, which is immodest, just like regarding the leg. The latter opinion holds that only regarding the leg we say that she would not remove it. If so, this has no connection to the argument in Sa'if 9.

Mishnah Berurah (51): Even if the string is decorative, it is not a Tachshit, for which we are concerned lest she remove it to show it off. It is needed for the garment. It is like a handle of the garment. Also, surely she would not remove it, lest her socks fall.

Mishnah Berurah (52): Shibolei ha'Leket and other Rishonim say 'a string or strap.' This is why it says 'we are not concerned lest they fall, i.e. the string or strap. This is clear from Semak.

Kaf ha'Chayim (77): R. Zalman says that we are not concerned lest the strap and socks fall, and she will carry them.

Shulchan Aruch (ibid.): She may go out with a bracelet on the arm or leg if it clings to the skin and does not fall off.

Gra (DH v'Yotz'ah): The Gemara asked only regarding Tum'ah, but not regarding the Heter to go out with it on Shabbos.

Gra (DH v'Hu): This is like the Rambam, who forbids Kevalim because they are loose. Biris and a bracelet cling, therefore they are permitted.

Mishnah Berurah (54): If it is loose, we are concerned lest it fall and she will come to carry it.

Shulchan Aruch (ibid.): Some forbid an arm bracelet.

Gra (DH v'Yesh): This opinion is like Rashi, who permits on her leg because she would not reveal her leg. This is not precise. Rashi forbids such a case even on the leg. He permits only on the socks.

Bi'ur Halachah (DH v'Yesh): The Magid Mishneh says that Rashi permits only a Biris. She would not remove it, lest her socks fall. I wrote that the Rambam discusses a bracelet directly on the arm . Rashi could agree to this (that she would not remove it)! However, the Gra argues with the Magid Mishneh oppositely. He says that Rashi permits only a Biris that holds up socks, but not a leg bracelet. She could remove it not in front of people and show it off!