Gemara

11b (Beraisa #1): A Zav (one who had emissions of uncooked semen) may not walk [in Reshus ha'Rabim on Shabbos] with a pouch [to catch emissions]. If he did, he is exempt;

12a - Question: The pouch will keep his [other] garments clean! (It should be considered carrying for this reason.)

Answer (R. Zeira): This Tana holds that anything worn to shield from filth is not considered carrying.

64b (Mishnah): A woman may go out with tufts in her ear [to absorb wax], in her sandal [for comfort] or [in the Ervah] to absorb Dam Nidah.

65a (Rami bar Chama): It must be tied between her thighs.

Rejection (Rava): It need not be tied. Since it is repulsive, we are not concerned lest she remove it.

92b (Mishnah): In truth, they said that a woman wearing [and carrying something in] a Sinar (short pants under her garment) is liable, because it normally goes back and forth.

Nedarim 55b (Beraisa): To shield oneself from rain, one may wear (in a Reshus ha'Rabim on Shabbos) Sak (a garment thick like sackcloth), a curtain, or a blanket, but he may not go with a box or basket or mattress;

Shepherds may go in Sak;

Anyone is allowed. Chachamim discussed a typical case.

Rishonim

Rif and Rosh (30b and 6:18): A Beraisa permits to go out with a garment thick like Sak, a curtain, or a blanket, due to rain. Shepherds, and anyone, may go in Sak. A Tosefta forbids with a box or basket or mattress.

Rambam (Hilchos Shabbos 19:17): One may go out with a garment thick like Sak, a curtain, or a thick blanket due to rain, but he may not go with a box or basket or mattress due to rain.

Tosfos (64b DH uv'Machshirei Ochel Nefesh): Rashi explains that the tuft is so that her garments will not get dirty. This is wrong. If so, it is to shield from filth, and it is a load, like it says on 12a! Rather, it is so the blood will not fall on her skin and dry up and pain her.

Beis Yosef (OC 301 DH v'Chen): Also the Ran, Mordechai, Semag and Hagahos Maimoniyos say like Tosfos.

Agur (433, cited in Beis Yosef OC 301 DH v'Chasav): My father says that coats that have Kefutzi (flaps?) in back, even though one wears them to shield from filth, one may go out with them on Shabbos if they are sewn. Then it is like a garment, for sometimes one wears them due to wind, cold and rain. Or Zaru'a says that R. Shimshon brings that the Yerushalmi permits this, but not if they hang from straps.

Beis Yosef (ibid.): l do not understand the difference between sewn and hanging. Since it hangs, there is no concern lest it fall and he will carry it. Perhaps when it is sewn it protects on the back of the neck from the cold, but not when it hangs. Then it is only to shield from filth, so it is forbidden. I say that we forbid (12a) something that is only to shield from filth only when it is not Derech Malbush, e.g. a Zav's pouch. We permit Derech Malbush even to shield from filth. It is not worse than towels that bathhouse attendants bring and hard felt that one may bring through wearing. One may wear Sak due to rain, even if he intends only to shield his garments from filth, since he covers himself Derech Malbush.

Poskim

Shulchan Aruch (OC 301:13): A Zav may not go out with a pouch to shield from emissions, to avoid getting dirty.

Mishnah Berurah (49): Similarly, one may not hang a cloth from his hat to clean his eyes.

Kaf ha'Chayim (74): Even though we exempt Melachah she'Einah Tzerichah l'Gufah, and we do not have Reshus ha'Rabim, it is forbidden.

Shulchan Aruch (ibid.): If a Nidah ties a garment in front of her so she will not get dirty from Dam Nidah, she may not go out with it, unless it is an undergarment that is [worn] like a garment.

Magen Avraham (20): If it is only in back of her and she ties it with straps in front of her, it is forbidden.

Magen Avraham (21): It says that she ties it, for if not, it would fall, for it hangs in front of her. Olas Shabbos erred.

Machatzis ha'Shekel: Olas Shabbos said that the Tur and Mechaber are unlike the conclusion on 65a, which permits even if it is not tied. He erred, for in Siman 303, the Tur and Mechaber permit even if it is not tied! Rather, here they discuss a typical case. If it were not tied, it would fall!

Gra (DH v'Chen): We learn about a Nidah from a Zav. This is unlike Rashi.

Gra (DH Sinar): A Sinar is Derech Malbush. This is clear from 92b.

Mishnah Berurah (50): It is a garment only if it is in front and in back, even if it has no bottom. If it is only in front or only in back and it is tied on, this is not a garment, and it is forbidden.

Kaf ha'Chayim (79): A Nidah is a bigger Chidush. Even though there is no concern lest it fall and she will carry it, it is forbidden, for it is a load.

Kaf ha'Chayim (80): Ezra decreed that women wear a Sinar (for modesty).

Shulchan Aruch (ibid.): If she ties it so the blood will not pain her, she may go out with it.

Mishnah Berurah (51): If she intends to avoid pain, even if it also shields from filth, it is Derech Malbush and it is permitted, just like other garments made to protect the body.

Bi'ur Halachah (DH Aval): A tuft inserted into her Ervah stays even if it is not tied. Here, it is in front of her, so it stays only if it is tied.

Kaf ha'Chayim (81): Also here, if it would stay without being tied, it is permitted.

Shulchan Aruch (14): One may go out on Shabbos with something Derech Malbush, even if he wears it only to save from filth.

Rema: Therefore one may wear a garment due to rain, or a hat on his head, but a woman may not put a garment on her kerchief due to rain, for this is not Derech Malbush.

Mishnah Berurah (53): This is even if it is an inferior garment, such as Sak.

Sha'ar ha'Tziyon (57): The Magen Avraham and Tosefes Shabbos say that Ituf (covering) suffices. It seems that only placing it on top of is forbidden, but covering the head suffices. The Pri Megadim connotes that the Magen Avraham obligates covering also part of the body. Perhaps he holds that the Magen Avraham agrees with the Taz. The Chayei Adam says that it must cover the majority of the body. I do not know his source. The Beis Yosef connotes unlike this.

Shulchan Aruch (21): One may not go with a box or basket or mattress.

Kaf ha'Chayim (115): Even if he puts it on Derech Malbush, it is forbidden, for it is not normal to wear such things. It is always a load.

Shulchan Aruch (ibid.): One may go out with Sak, a curtain, or a blanket (thick garments).

Beis Yosef (DH Ein): Perush ha'Rosh (Nedarim 55b DH Yotz'im) says that one may go out with Sak, and it is not a load, for it is normal to go out with due to rain. It is permitted even not in the rain. One may not go out with a box, for this is not Derech Malbush. It is a load. This is unlike the Roke'ach (100), who permits going out with Sak only due to rain.

Gra (DH Ein): The Rosh permits Sak even not in the rain. The Beraisa mentioned in the rain for a Chidush, like it says in Sa'if 14.

Damesek Eliezer: I.e. in the rain is to shield from filth, which is a reason to forbid. Even so, thick Sak is permitted, for it is Derech Malbush.

Mishnah Berurah (72): Since shepherds often go out with Sak due to rain, since it is a garment for them, it is permitted for everyone, even not due to rain.

Kaf ha'Chayim (116): The Mechaber ruled Stam like the Rosh, who permits even not in the rain, and so rule the Magen Avraham, Tosefes Shabbos and R. Zalman. Olas Shabbos and Eliyahu Rabah say to be stringent like the Rambam, and wear them only in the rain.