Gemara

87b (Mishnah): If a wall between two Chatzeros is 10 Tefachim tall and four Tefachim thick, the Chatzeros can be Me'arev individually, but not together;

If there are Peros on top of the wall, people from either Chatzer may alight on the wall and eat them, but they may not take them down.

91a - Question (Rav Sheshes - Mishnah - R. Shimon): Roofs, Chatzeros and Karfifos are all considered one Reshus for Kelim that were Shoves in them, but not for Kelim that were Shoves in the house.

If Rav permits even when they were not Me'arev, Kelim that were Shoves in the house will not be found in the Chatzer! (Does he teach about one who transgressed and carried them out to the Chatzer?!)

Answer (Rav Sheshes): He teaches about hats and scarves.

Shabbos 147a: Ratanin (couriers) may go out with scarves folded on their shoulders;

The same applies to anyone, just it is normal for couriers to go like this.

R. Yehudah: A case occurred in which Hurkenus ben R. Eliezer went out with a scarf folded on his shoulder. A thread from it was wrapped on his finger;

Chachamim said that it is permitted even without a thread wrapped on his finger.

(Rav Nachman bar Rav Chisda): The Halachah is, it is permitted even without a thread wrapped on his finger.

147b (R. Chiya bar Aba): Bathhouse attendants may bring towels for women to the bathhouse [through Reshus ha'Rabim, via wearing them], on condition that they cover the majority of their heads and bodies with them.

To wear Sachnisa (this will be explained) in Reshus ha'Rabim, one must tie the ends at the bottom;

(R. Chiya bar Aba): One must tie it below the shoulders.

Rishonim

Rambam (Hilchos Shabbos 19:19): One may go with a scarf on his shoulder, even if it is not tied to his finger through a string. One may not go with a scarf that does not cover Rosho v'Rubo. If it was a short Sachnis (scarf) that is not wide, he ties the two ends below the shoulders, and it is like a belt, and he may go out with it.

Rashi (91a DH b'Kumsa): The hats and scarves were in the house at the start of Shabbos. He took them into the Chatzer Derech Malbush (through wearing them).

R. Akiva Eiger (91a DH b'Kumsa): Why does Rashi (Shabbos 130b DH Mutar) say that if one wore something from the house to a Mavoy [without an Eruv], he may carry it in the entire Mavoy? Our Gemara forbids taking it to another Chatzer! The Ritva and Rashba explicitly forbid; 87b proves that they are correct. Rashi (there) requires investigation.

Poskim

Shulchan Aruch (OC 301:34): One may wear a scarf folded on his shoulder, even if there is not a thread tied to his finger.

Beis Yosef (DH Yotzei): The Gemara permits even if it is not tied. The Rambam rules like this. The only opinion I saw unlike this is R. Yerucham in the name of R. Chananel, who requires a thread tied to his finger, and says that this is the custom. I do not understand why, unless he had a different text.

Beis Yosef (DH v'Chasav): R. Yerucham says that one may go out wrapped in a scarf like a jacket. This connotes that it is permitted only this way. This is difficult, for "one may go out with a scarf folded on his shoulder" connotes even if he is not wrapped in it! This is unlike a Talis, for a scarf is normally worn this way. The Rambam connotes like this. He obligates one who goes out with a Talis folded on his shoulder, but permits this regarding a scarf. The Rashba (5:224) was asked why the custom is to wear a folded Talis. He answered that Bnei Sefard go out this way for modesty, even in the Chatzer or house. It is Derech Malbush for them. It is like we say that couriers go out with a scarf on their shoulders.

Taz (24): Why is a folded scarf permitted more than a folded Talis? If they have different forms, the Poskim should have discussed this, like the Tur did regarding Sachnisa! I explain the Sugya differently. Rashi said that after the Talis was on his head, he picked up the ends and put them on his shoulders. Had he found it folded, this would be Derech Malbush. There is no difference between a Talis and a scarf. The Tur holds like this, just he cited the Gemara, in which R. Yochanan forbids a Talis [that he folded after putting it on], and the Beraisa discusses a scarf that was already folded. The Rambam permits a scarf only if it covers Rosho v'Rubo. He learns from bathhouse attendants. There, we do not discuss folding! He explains R. Yochanan simply, unlike Rashi, that he put on a folded Talis. Even though this is the way of Ituf (wrapping), it is forbidden. He permits a scarf as long as it covers Rosho v'Rubo. The Rambam did not say 'on his shoulder' to exclude the rest of the body. Rather, the majority suffices. This is like R. Yerucham! A proper garment is permitted as long as he did not fold it after putting it on. What is not a proper garment, e.g. a towel, is forbidden even without folding, unless it covers Rosho v'Rubo.

Kaf ha'Chayim (195): Eliyahu Rabah and the Acharonim were not concerned for the Taz' questions. They rule like the Shulchan Aruch.

Gra (DH b'Sudar): The Aruch and Rambam explain that Ratanin are couriers. They go out with folded scarves, lest it interfere with running.

Mishnah Berurah (127): We discuss places where some people go out like this sometimes.

Shulchan Aruch (ibid.): If the scarf does not cover Rosho v'Rubo, it is forbidden unless he ties the two ends together below his shoulder.

Beis Yosef (DH v'Chasav ha'Rambam): The Rambam forbids going with a scarf that does not cover Rosho v'Rubo. It seems that he learns from bathhouse attendants. They may bring towels only if they cover Rosho v'Rubo. One must tie the two ends of Savnisa below the shoulders. He explains "only if they cover Rosho v'Rubo", i.e. the towels must be wide enough to cover Rosho v'Rubo. If it is wide enough, it is permitted even if he did not cover Rosho v'Rubo, e.g. it is on his shoulder. He holds that Savnisa is short and not wide. Therefore, it is forbidden unless he ties the two ends below the shoulders. Rashi says that it is big, and hangs between his shoulders, and his head is wrapped in it. He explains that one must cover Rosho v'Rubo in it, for if not, it is not Derech Malbush, unless he ties the two ends below his shoulders.

Bi'ur Halachah (DH v'Im): The Taz says that the Rambam always requires covering Rosho v'Rubo. I distinguish. We cannot be stringent like the Taz about a scarf folded on his shoulders, since many Rishonim are lenient. If the custom spread to cover Rosho v'Rubo, only this is Derech Malbush. The Rambam does not contradict the general rule I wrote at the beginning of this Sa'if, that a folded scarf on his shoulders is permitted.

Kaf ha'Chayim (198): The Beis Yosef says that if it does not cover Rosho v'Rubo, it is not Derech Malbush. The Levush says that since it is so small, it is prone to fall if it is not tied.

Kaf ha'Chayim (199): Here, the custom is that the scarf covers only the body, so this suffices.

Magen Avraham (42): Some Gedolim forbid the way Chachamim hang a scarf on the shoulders for a Tachshit. In Konstantina they are not careful. Meticulous people tuck the ends of the scarf into their belts, and then there is no concern lest they remove it. I say that they must tie it in.

Kaf ha'Chayim (196): Since it is a Tachshit for a Chacham, it is not a load, just we are concerned lest one take it off to show it off.

Mishnah Berurah (133): One may tie a handkerchief around the neck Derech Malbush.

Kaf ha'Chayim (132): Some wrap a handkerchief around their neck. Where this is not the custom, it is not considered Derech Malbush. However, sometimes people do so due to the cold, so perhaps we need not protest if it is Derech Malbush. If they wrap it once and the ends hang down, this is totally Asur. They will not heed us; it is better that they be Shogeg. In any case one should tell them to wrap it around the neck.

Ohr l'Tziyon (2:23:6,12): If one wraps the handkerchief around the neck, this is not considered a garment. This is carrying through a Shinuy. This is forbidden only mid'Rabanan, so one may rely on an Eruv for this.

Ohr l'Tziyon (2:23:7): One may wear a coat on the shoulders without inserting the arms in the sleeves, and go out in Reshus ha'Rabim. This is Derech Malbush, just like a scarf folded on his shoulders, which the Beis Yosef permits.