Gemara

(Beraisa): One may dry himself with a towel and leave it in a [nearby] window. He may not give it to a bathhouse attendant, for they are suspected about this (squeezing them);

R. Shimon says, he may even take it home.

91a (Beraisa - Rebbi): When we learned from R. Shimon, we would bring to him [on Shabbos] oil and a towel, transferring them from Chatzer to roof, from roof to Karfef, and from Karfef to Karfef, until we reached the bathing spring.

Shabbos 147a (R. Yitzchak bar Yosef): If one walks in Reshus ha'Rabim on Shabbos with his garment folded onto his shoulder, he is Chayav Chatas.

Support (Beraisa): If a clothes salesman walks in Reshus ha'Rabim on Shabbos with a garment folded onto his shoulder, he is Chayav Chatas;

The same applies to anyone, just clothes salesmen normally go like this.

(Ula): R. Ilai forbids Marzev on Shabbos.

Question: What is Marzev?

Answer (R. Zeira): People of Bavel make folds in garments that are too long.

Rav Dimi: Once, Rebbi entered a field. The two sides of his Talis were on his shoulders;

He heard that R. Meir (according to one version - R. Akiva) is Mechayev Chatas for this. He was surprised, but lowered his Talis.

Rav Shmuel bar Rav Yehudah: Rebbi was asked about wearing a Talis this way. He was about to permit, until he heard that R. Meir or R. Akiva is Mechayev.

147b (R. Chiya bar Aba citing R. Yochanan): A bathhouse attendant may bring towels for women to the bathhouse, on condition that he covers the majority of his head and body with it.

Rava (to people of his city): When [Nochri] soldiers force you to take their clothes to the bathhouse [on Shabbos], let them down from your shoulders [so it will be Derech Malbush, i.e. the way of wearing them].

Rishonim

Rambam (Hilchos Shabbos 19:18): If one wraps himself in his Talis and folded it on both sides in his hand or on his shoulder, if he intended to gather the corners so they will not tear or get dirty it is forbidden. If he gathered them to adorn himself, like the local custom, it is permitted.

Rambam (19): (R. Yitzchak bar Yosef): If one walks in Reshus ha'Rabim on Shabbos with his garment folded onto his shoulder he is liable.

Rambam (22:20): One who bathed in water may dry himself with a towel and bring it. We are not concerned lest one squeeze.

Rashi (147b DH Ad): The Beraisa proves that one may take back the towel after drying with it, for presumably they did not leave the towel in a Hefker place.

Rosh (Shabbos 22:11): Some say that Marzev is folding two corners onto each shoulder. The pockets of Bavel are similar. It is a question of Hotza'ah on Shabbos, for this is not Derech Malbush.

Me'iri (147a DH ha'Ulyarin, citing the Ra'avad): A bathhouse attendant may bring a towel only if he covers Rosho v'Rubo with it, for then it is Derech Malbush. Normally we do not require Rosho v'Rubo for towels. Here we are stringent, since he brings it for the sake of others.

Poskim

Shulchan Aruch (OC 301:29): If one goes out with a Talis folded on his shoulders, i.e. after putting it on his head, he lifts the bottom onto his shoulders, he is Chayav Chatas.

Beis Yosef (DH ha'Yotzei): Rashi says that one who goes out with a Talis folded on his shoulders, i.e. after putting it on his head, he lifts the bottom onto his shoulders, is Chayav Chatas, for it is not Derech Malbush. The Ran (62a DH ha'Yotzei) says that this is unlike the folded Talis below, in which the two sides of the Talis in front and in back rest on his shoulders. In Rebbi's case, only the sides in front were folded. Therefore, he thought that it is permitted until he heard differently. From here, we must warn Chasanim not to go out to Reshus ha'Rabim when the sides of the Talis are folded on their shoulders. Also R. Yonah says so.

Taz (18): Since he is wrapped with the Talis until under his shoulders, it is permitted even though he lifts up the bottom and folds it.

Mishnah Berurah (110): All the more so, if he totally removed it, folded it and placed it on his shoulder, he is liable.

Shulchan Aruch (ibid.): If it hangs down widthwise from his shoulders, it is permitted. Since he wraps himself in it and covers with it his shoulders and body, even though it narrows a little below, it is permitted.

Beis Yosef (DH Mihu): If it is not folded on his shoulders, rather, hanging down widthwise below the shoulders, it is permitted. This is like the bathhouse attendants. Rashi explains that the bottom should not rest on his shoulders, rather, extended, so it looks like a garment. Shibolei ha'Leket brings from R. Tam that "going out with a folded Talis resting on his shoulders" means that after putting it on his head, he lifted the bottom on his shoulders. One may wrap himself with his Talis, even though it narrows a little below, [since] he covers his shoulders and body. A Gaon forbids folding a Talis on his head and shoulders on Shabbos to protect from the sun or rain. According to R. Chananel, it is forbidden only if he totally removes it, and folds and carries it on his shoulder.

Beis Yosef (DH v'Al): Rashi explains that Marzev is folding garments that are too long. It is forbidden, for it is like fixing the garment. Tosfos, the Rosh and R. Yerucham explain that it is due to Hotza'ah, for this is not Derech Malbush. It seems that the Rambam agrees. Tosfos forbids when the two right corners are folded on the left shoulder, so that he can walk quickly and the Talis will not impede him. If it is for adornment, it is fine, i.e. when the sides of the Talis are not gathered. Rather, one side rests on his shoulder by the neck, and the other falls on the shoulder until the lower arm. The Aruch, Rosh and R. Yerucham say so.

Beis Yosef (DH veha'Kolbo): The Kolbo says that the Ra'avad forbids a folded Talis only when he folds the right side on the right, and the left side on the left. If he folds the left side on the right, and the right side on the left, like we do, it is permitted. This is for adornment. All the more so, Chasanim may go out with it hanging down on one side.

Gra (DH Aval): If the sides are not on his shoulders, surely it is permitted.

Mishnah Berurah (112): This is even if it does not cover the majority of his body. Some are stringent and require that it covers the majority of his body.

Kaf ha'Chayim (170): This is when it is only on his shoulder. If even a little is on his body below his shoulders, this is Derech Malbush. R. Zalman requires that it covers the majority of the body, and the Taz connotes like this.

Shulchan Aruch (ibid.): Based on this, one may wrap himself with a Talis under his cloak in order to bring it to the Beis ha'Keneses.

Mishnah Berurah (113): I.e. even though it gets slightly narrower below, since he is wrapped until below the shoulder, we permit even if he lifts the bottom. It is difficult to be wrapped like in the Beis ha'Keneses when walking in the street, even if he has a cloak over it.

Shulchan Aruch (34): One may wear a scarf folded on his shoulder. If it does not cover Rosho v'Rubo, he must tie the two ends together below his shoulder.

Shulchan Aruch (48): One may dry himself with a towel and bring it in his hand.

Mishnah Berurah (175): He may bring it home where there is an Eruv.

Shmiras Shabbos k'Hilchasah (14:25): One may wrap himself with a towel and go out with it where there is no Eruv.

Ashrei ha'Ish (2:14:7): One may wrap a towel around his neck like a scarf and go out with it. Even so, it is exempt from Tzitzis, for it was not made to be worn.

Ohr l'Tziyon (2:23:8): One may put a towel on his shoulders and go out to Reshus ha'Rabim. This is Derech Malbush, just like a scarf on his shoulders. Even so, one need not attach Tzitzis. Even those who are stringent about this during the week permit on Shabbos. Since one cannot affix Tzitzis on Shabbos, there is no obligation. In OC 13:3, if one found out in Karmelis that [the Tzitziyos on] his Talis is (are) invalid, he may wear it home, due to Kevod ha'Briyos. It seems that this is only if he found out on Shabbos, but not if he knew before Shabbos, for then he was obligated to attach Tzitzis. A towel is more lenient, for he does not intend for Malbush, and especially if it is not of wool, for we hold like the Rambam that the Torah does not obligate putting Tzitzis on it.

Rebuttal (Ha'aros on Birkei Yosef, 1): This is not Derech Malbush during the week, and not on Shabbos. Also, the towel is not big enough to wrap the majority of the body, which we require (Sa'if 34). Also, a towel around the neck like a scarf is never Derech Malbush.