Gemara

11b (Beraisa): 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.

59b - Question (Ravina): May one wear Kamra (a belt with gold and gems woven inside, or it is all gold) and also a belt?

Answer (Rav Ashi): This is [forbidden,] like wearing two belts.

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

Mordechai (464, Daf 11): A woman may not go out to the Beis ha'Keneses in winter with a cover on her head lest rain descend on her kerchief, for this is [like] saving from filth.

Tosfos (59b DH Trei): Two belts, one on top of the other, are forbidden. This is unlike wearing many garments due to the cold. Here, there is no benefit from a second belt. It is not normal to do so. It is a load.

Poskim

Shulchan Aruch (OC 301:14): One may go out on Shabbos with something Derech Malbush, even if he wears it only to save from filth.

Beis Yosef (DH Kasuv): The Mordechai forbids a cover on her head lest rain descend on her kerchief, for this is saving from filth. Based on this, the Beraisa that permits Sak due to rain must discuss when he does not wear it to save his clothing [from getting wet], rather, to save his body from harm from the water.

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.

Beis Yosef (DH v'Chasav): The Mordechai forbids a cover to protect a kerchief from garment, i.e. when it is not Derech Malbush. Perhaps this is the source for the custom for men to put a big felt hat over the (regular) hat due to rain. We do not forbid lest it fall and he carry it, like Rashi (138b) explained regarding Si'ana (a felt hat). Since it is deep and enters the hat, there is no concern.

Taz (10): This is when it covers only the kerchief. If it covers also the body, this is Derech Malbush and it is permitted, just like bathhouse attendants who bring towels for women to dry themselves.

Mishnah Berurah (54): This is only if it is tied well [so it will not fall].

Magen Avraham (23): Maharil says that women put a piece of linen fabric on the kerchief when rain falls to shield from filth of the rain. Also rich women may do so [even] in Reshus ha'Rabim, since it is considered a garment for poor women.

Mishnah Berurah (52): The Mechaber discusses what he wears above his garments. The Rema adds "a garment due to rain" to teach that even if he wears a garment above his garment just due to rain, and if not for the rain, he would not wear it, e.g. it is thick and coarse like Sak. Similarly, the Rema discusses a hat worn above the big hat on his head, so the hat will not get dirty. Even so, it is permitted because it is Derech Malbush.

Mishnah Berurah (55): Therefore, a man may not cover his hat with Patshila (a handkerchief).

Kaf ha'Chayim (84): Therefore, one may wear rubbers (or boots) over his shoes.

Shulchan Aruch (36): One may go out on Shabbos with two shirts or two coats one on top of the other, whether for his own need or for the need of his friend.

Beis Yosef (DH uv'Kolbo): Tosfos, the Rosh, Mordechai, Semag, Semak and Sefer ha'Terumah permit two coats. However, the Mordechai says that R. Yo'el forbids two cloaks one on top of the other for the sake of his friend. This is not so clear to me. In any case, since many permit, we follow them.

Beis Yosef (DH u'Mah): The Tur brings that Sefer ha'Mitzvos forbids two cloaks. I did not see anyone say so. The reason is that there is no benefit from the second cloak and it is not normal to wear it, so it is a load, like Tosfos and the Rosh say about two belts. However, Tosfos (147b DH Tzarich) says that Sanbita is a very thin garment. One must tie the ends below, so the wind will not blow it, or because it looks like a load and not like a garment, unless it is tied. Tosfos permits even though it is very thin and he does not benefit from it, since he carries it Derech Malbush (in the way of wearing).

Rema: One may wear two hats one on top of the other.

Beis Yosef (DH Kasav ha'Agur): The Agur and Shibolei ha'Leket permit going out with two [pairs of] socks or two turbans due to the cold.

Beis Yosef (DH Kasuv b'Shibolei): Shibolei ha'Leket, citing the Yere'im, forbids wearing two coats or other ornaments one on top of the other, and he needs only one of them. Regarding a fire, one may cover himself and wear whatever he can. This implies that not in place of a fire, it is forbidden. Sefer ha'Terumah permits to go out with two coats due to the cold.

Beis Yosef (DH Kasav ha'Agur): The Agur and Shibolei ha'Leket permit going out with two [pairs of] socks or two turbans due to the cold.

Mishnah Berurah (136): During the week, it is normal to wear a big hat over a small hat.

Kaf ha'Chayim (213): This refers to a hat over a Kipah. No one normally wears a second hat over a proper hat.

Kaf ha'Chayim (215): Under one's hat, one may wear a small hat to absorb perspiration.

Shmiras Shabbos k'Hilchasah (18:10 and footnote 46, citing ha'Gaon R. S. Z. Auerbach, Ztz"l): One may wear a rain hat over his hat. It is made to fit the hat nicely, so people are not particular to remove it once the rain stops. It is Batel to the hat. It is considered Derech Malbush.

Igros Moshe (1 OC 108, b'Sof): Nedarim 55b permits going out with thick Sak. We cannot prove from there that one may go out with a plastic cover over a hat. The Sak is worn Derech Malbush. Therefore it is permitted even to shield from filth. The Shulchan Aruch (Sa'if 13) says that if a Nidah ties a garment to shield from filth, it is permitted only if it is Derech Malbush. The Taz and Magen Avraham say similarly about a garment over a kerchief due to rain. It is Derech Malbush only if it covers also part of her body, and not just the kerchief. The same applies to men. If the hat cover is only on the hat, it is unlike Sak, which is at least partially on the body. The Beis Yosef says that the custom is for men to wear a felt cover on hats is because it is Derech Malbush. This is why the Rema (Sa'if 14) said on his head, even though he discusses a cover over the hat, for it must also cover part of the head.

Igros Moshe (ibid., DH u'Mah): The Rema (Sa'if 36) permits two hats that are normally worn together for Noy (beautiy or adornment) or warmth. This is unlike two belts, which people do not wear. We are not concerned for one who wears two belts; the Halachah is based on normal people. Surely people would not wear two hats if there were no need for this. I say that if the second hat is for warmth, it is permitted even if it is not normal to wear it. We do not say that Batlah Da'ato Etzel Kol Adam, for wearing is primarily for warmth. However, a plastic rain hat gives no warmth, and it is not Noy. It detracts from the Noy of the hat! It is only to shield from filth, so it is forbidden. Boots or rubbers that are worn over shoes are different. They also protect the body from pain. When the shoes get wet, the feet feel the wetness and cold. A Nidah may go out with a tuft to save her from pain.

Minchas Yitzchak (3:26:8:1): Here, the custom is that women cover [their kerchiefs] at a time of rain [but men do not cover their hats]. The plastic covers the kerchief, ears and part of the face. Straps tie it under the jaw. It covers the entire head. There are two reasons to permit. Firstly, the Magen Avraham (23) permits Derech Malbush even if it is to shield from filth. It seems that this is even if it covers only the head, like Sha'ar ha'Tziyun explains, unlike the Pri Megadim. Secondly, it is also to shield her from pain due to rain, i.e. exposed parts of her head. Also, if there is much rain, it will penetrate through the kerchief, and she might get chilled. Avodas ha'Gershoni permits covers to protect the kerchief, for they also prevent rain from dripping on the back of the neck. It is not a load, and we are not concerned lest she remove it to show off the beautiful kerchief. In any case one who is lenient does not lose.

Minchas Yitzchak (2): He (Avodas ha'Gershoni) did not explain why it is not a load. It seems that it is because she covers herself in it, and it shields from bodily pain. However, in practice I am stringent. Avodas ha'Gershoni was not concerned lest she remove it when the rain stops to show off her beautiful kerchief, and carry the cover four Amos in Reshus ha'Rabim, because normally weekday clothes are not so esteemed to show them off. When she removes the cover to show off the kerchief, this will remind her that it is Shabbos, and she will not carry in a Reshus ha'Rabim. Regarding a plastic cover, which people normally wear only at a time of rain, it is normal to remove it once the rain stops, for it is thin and folds into a very small space. There is concern lest one remove it to avoid being mocked. We can ask similarly regarding Sak that is worn only at a time of rain.

Ashrei ha'Ish (citing Kisvei Avnei Yashfe, in the name of ha'Gaon R. Y.S. Elyashiv Ztz"l): One may not go out with a plastic bag over a hat to shield from rain, for it is not a Malbush. It is permitted only if it also comes down on the face or ears.

Yabi'a Omer (5 OC 24): There is no concern for an Isur Torah regarding a rain hat, since we do not have Reshus ha'Rabim, and even if there is Reshus ha'Rabim in big cities in America, it is a Melachah she'Einah Tzerichah l'Gufah. We may rely on comparisons to permit. Regarding men, we do not decree lest one [remove it and] carry in Reshus ha'Rabim. If we will be stringent, people will refrain from coming to the Beis ha'Keneses or Beis Midrash due to concern for ruining their hats. Also Bris Olam (ha'Gaon R. Y. Zilber Shlita) says so.