Gemara

(Rava): R. Akiva says (in the Mishnah) gives a general rule. Only those things that cannot be done before Shabbos are Docheh Shabbos.

37b (Rav Huna): If invalid Tzitzis are on a garment, one who enters (or walks four Amos in) Reshus ha'Rabim while wearing it brings a Chatas (if he was Shogeg).

Version #1: On Shabbos, Ravina noticed that a corner of Mar bar Rav Ashi's garment (with the Tzitzis) had snapped off. He did not tell him until Mar bar Rav Ashi reached his house.

Mar bar Rav Ashi: Had you told me earlier, I would have taken off my garment right away!

2. Ravina: That would be embarrassing (to walk home without your garment). Kavod ha'Briyos (avoiding disgrace) overrides a Lav (walking in Reshus ha'Rabim wearing a garment with invalid Tzitzis)!

3. Mar bar Rav Ashi: Rav bar Sheva explained that it overrides only the Lav of Lo Sasur (which commands us to follow mid'Rabanan laws).

Version #2: Ravina immediately told Mar bar Rav Ashi that a corner of his garment had snapped off.

Mar bar Rav Ashi: Do you think I must take off my garment right away? Kavod ha'Briyos overrides a Lav!

Question: Rav bar Sheva explained that it overrides only the Lav of Lo Sasur!

Answer: Mar bar Rav Ashi was not walking in Reshus ha'Rabim, rather, in a Karmelis, where the Isur to carry is only mid'Rabanan.

Shabbos 131a (Beraisa - R. Eliezer): Lulav (and Sukah, Matzah, and Shofar) override(s) Shabbos.

(Beraisa): All agree that if one put Tzitzis on his garment on Shabbos, he is liable.

131b (Rav Nachman): He could make his garment Hefker. (Therefore, the Mitzvah is not obligatory. All agree that it does not override Shabbos.)

139b (R. Avin bar Rav Huna): One may cloak himself in a Kilah (a canopy over a bed) and walk in Reshus ha'Rabim, without concern for the straps (that hang from it).

Question: Rav Huna taught that if one walks in Reshus ha'Rabim on Shabbos with Tzitzis that do not fulfill the Mitzvah, he is Chayav Chatas (they are like a load)!

Answer: (Even invalid) Tzitzis are esteemed, so they are not Batul to the garment. Straps are Batul to the Kilah.

Berachos 19b (Beraisa): If one finds Sha'atnez in his garment, even in the market, he must take it off.

Question (Beraisa) "Ul'Achoso" teaches that a Kohen Gadol is not Metamei for relatives, but he is Metamei for a Mes Mitzvah, even if he is also a Nazir, and he will lose the Mitzvah of Korban Pesach or circumcising his son today due to this.

Answer: There is different, for it says "ul'Achoso."

Question: We should learn from here that Kavod ha'Beriyos overrides every Lav mid'Oraisa!

Answer: This applies only to a passive transgression. (Wearing Sha'atnez is considered an action.)

Rishonim

Rambam (Hilchos Shabbos 19:20): If a Talis has Kosher Tzitzis, one may walk (in Reshus ha'Rabim on Shabbos) while wearing it during the day or at night. The Tzitzis are not a load. They are Noy (decoration) for the garment, and part of its design, like a hem.

Teshuvas ha'Rosh (2:9): The Ri explains that Tzitzis are not Batel because he intends to fix the snapped ones and join them to the intact ones.

Mordechai (Hilchos Ketanos 944): R. Shlomo of Drush said that if a thread of one's Tzitzis snapped on Shabbos, one may not wear it until he fixes it, for he transgresses the Aseh (of Tzitzis). The Ri rejected this from our Sugya. Even though it was in a Karmelis, he continued wearing it, due to Kavod ha'Briyos. This overrides only Lo Sasur (but not a Torah Lav or Aseh). Rather, the Aseh of Tzitzis is to put on Tzitzis when wearing it. No Lav forbids wearing a garment without Tzitzis. One transgresses the Aseh on a weekday, but not on Shabbos, when he cannot put on Tzitzis. The same applies to (a house without a) Mezuzah or railing around the roof. Surely, we do not forbid to enter a house without a Mezuzah!

Poskim

Shulchan Aruch (OC 13:3): If on Shabbos one was in a Karmelis, and he found out that the Talis he is wearing is Pasul, he does not remove it until reaching his house, for Kavod ha'Briyos is great.

Beis Yosef (DH Garsinan): It seems that even if one is in Karmelis and realizes that his Tzitzis were Pasul when he left (his house), we do not fine him for leaving without checking. We do not fine when this impinges on Kavod ha'Briyos, especially since we do not find that Chachamim obligated him to check. Perhaps the Gemara is lenient only for their Talesim, which they wore in place of a jacket. It would be embarrassing to remove it in the market. We do not wear Talesim for the sake of clothing. It is not embarrassing to remove it, so one removes it even in Karmelis. Perhaps even for us it is embarrassing. Alternatively, since it was embarrassing for them, Chachamim never decreed in Karmelis. They did not distinguish different kinds of Talesim. This is why the Magid Mishneh and Nimukei Yosef brought this law Stam, without distinguishing.

Mishnah Berurah (10): R. Akiva Eiger was skeptical about saying that Chachamim did not distinguish between kinds of Takesim.

Mishnah Berurah (8): Even if he knew before Shabbos, but forgot to put on Tzitzis, we do not fine him. R. Akiva Eiger proves this from one who forgot to make an Eruv (Rema 355:3).

Rema: He need not remove even the Talis Katan under his garments. The same applies if one of his Tzitzis snapped and he is embarrassed to sit without a Talis. He may wear it without a Berachah, due to Kavod ha'Briyos. This is only on Shabbos, when he cannot make Tzitzis. On a weekday, such a case is forbidden.

Beis Yosef (DH Kasuv b'Mordechai): Teshuvah Maharam (287) says that if one went to the Beis ha'Keneses and found that his Tzitzis snapped and it is Pasul, and he is embarrassed to change his custom and sit in front of the Tzibur without a Talis, perhaps he may wear it without a Berachah. We find in Menachos that Kavod ha'Briyos permits in Karmelis. I disagree. Sitting in front of the Tzibur without a Talis is unlike removing one's Talis in the market.

Taz (5): According to the Ri brought in the Mordechai, the Heter is not due to Kavod ha'Briyos. It is always forbidden on weekdays, but permitted on Shabbos, when he cannot fix it. Perhaps the Rema permits only for Kavod ha'Briyos because the Mordechai brought another opinion. He permits only on Shabbos, because sitting without a Talis is not as embarrassing as removing one's Talis in the market, like the Beis Yosef said. In Menachos we needed Kavod ha'Briyos to permit in Karmelis. The Ri totally permits on Shabbos in Reshus ha'Yachid.

Magen Avraham (7): Even though it is not embarrassing to go without a Talis Katan, removing it is embarrassing.

Magen Avraham (8): Even though one does not transgress an Aseh on Shabbos, surely it is forbidden mid'Rabanan. We permit only due to Kavod ha'Briyos. This is unlike Olas Tamid. The Mordechai says that there is no Isur on Shabbos. In Shabbos (131b), we say that if Tzitzis is a Chiyuv of the garment (even without wearing it), one who keeps it in his house without Tzitzis transgresses, even on Shabbos. If so, according to the opinion that Tzitzis is a Chiyuv of the person (who wears it), why it is permitted mid'Oraisa to wear it on Shabbos without Tzitzis? We must say that this is according to R. Eliezer, who holds that Tzitzis overrides Shabbos, so the Aseh applies even on Shabbos. We hold that Tzitzis does not override Shabbos. Seemingly, it should be permitted to wear it even on a weekday! Kavod ha'Briyos overrides an Aseh of the Torah if one is only passive (Berachos 20a)! It is clear from there that wearing a garment without Tzitzis is considered passive. Also Tosfos (Yevamos 90b DH Kulhu) says so.

Machatzis ha'Shekel (DH v'Chen): Yevamos 90b itself says so. Tosfos explains the Gemara, that Chachamim were able to exempt a linen garment from Tzitzis (lest one attach wool Tzitzis and transgress Sha'atnez) because there is no Isur to put the garment on, and afterwards one transgresses passively.

Magen Avraham (ibid.): The Mordechai mentioned Shabbos only at the beginning, when he said that R. Shlomo forbids even on Shabbos.

Machatzis ha'Shekel (DH v'Gam): He means that the Mordechai mentioned Shabbos only because he initially cited R. Shlomo, who forbids even on Shabbos.

Magen Avraham (ibid.): We can say that the Mordechai permits even on a weekday. Even though one transgresses an Aseh, perhaps he need not remove it due to Kavod ha'Briyos! He brought proofs from Mezuzah and a railing.

Machatzis ha'Shekel (DH she'Haei): Surely, even on a weekday one may enter a house without a Mezuzah or railing.

Magen Avraham (ibid.): The Beis Yosef brings from Maharam to permit even on a weekday. We must say that the source (for the Shulchan Aruch and Rema to forbid on a weekday) is Piskei Tosfos (Menachos 130) and the Agur. This is like one answer in Tosfos Shevu'os (30b DH Kulhu), that Chachamim permitted transgressing passively only for a great disgrace. A Chacham must go to Beis Din to testify about an Isur (CM 28:5), even though it is disgraceful for him. They hold that sitting without a Talis is a small disgrace. I say that if Tzitzis of a Talis Katan snapped in Reshus ha'Rabim, even on a weekday he need not remove it, for removing it is a great disgrace. The Shulchan Aruch discusses Shabbos, for then he need not rush to go home to take it off. On a weekday, he must rush. One who sees that another's Tzitzis snapped in Reshus ha'Rabim, he should not tell him until he gets home. If Tzitzis are not available in the city on a weekday, this is like Shabbos. Why don't we obligate him to give his Talis to another for a gift, and then borrow it, so it will be exempt? In any case it is obligated mid'Rabanan, for it looks like his, like we say about a Talis borrowed for more than 30 days (OC 14:3). I answer that if we would say so on Shabbos, even on a weekday he will do so to exempt himself. Also, one may give a gift on Shabbos only for needs of Shabbos.

Gra (DH v'Hu): The Gemara said that in a Karmelis there is only an Isur mid'Rabanan. This shows that there is no Isur Aseh mid'Oraisa on Shabbos, because one cannot attach Tzitzis on Shabbos.

Mishnah Berurah (12,14): This Heter is only if he is embarrassed to sit without a Talis, and he cannot borrow a Talis in the Beis ha'Keneses.

Mishnah Berurah (15): Nowadays, taking off a Talis Gadol in Reshus ha'Rabim is a small disgrace. If after putting on the Talis he sees that one of the Tzitzis is Pasul, many Acharonim do not require removing it immediately. Doing so in front of many is a great disgrace. However, he must rush home or to a house in front of the Beis ha'Keneses and remove it. The Chayei Adam and Pri Megadim are stringent. They say that even this is a small disgrace, for many take off their Talis in the Beis ha'Keneses. Therefore, one should do like Orchos Chayim says: he makes it Hefker, and then it is exempt from Tzitzis. When he gets home, he acquires it again.

Note: How can Orchos Chayim explain Version #1, in which Ravina did not tell Mar bar Rav Ashi until Mar reached his house? He should have told him, so he will make it Hefker!

Mishnah Berurah (16): As long as one is engaged in a Mitzvah in the Beis ha'Keneses, on Shabbos he need not leave to take off the Talis. On a weekday he must leave immediately, for there is an Isur Torah to leave it on. If one sees that another's Tzitzis are Pasul, he should call him to his house, and then tell him. A Talis obligated mid'Rabanan, e.g. it was borrowed more than 30 days, has the law of a Talis on Shabbos.

Mishnah Berurah (9:3): If garments of materials other than wool and linen are obligated only mid'Rabanan, if the Tzitzis snapped on Shabbos, Kavod ha'Briyos permits contuinuing to weasr it.

Kaf ha'Chayim (10): Shirei Keneses ha'Gedolah says that the Rema: permits only if he did not yet put on a Talis, for then people think that he does not have a Kosher Talis and it is embarrassing. If he put it on and then it became Pasul, this is not embarrassing. Most say that the Rema permits in both cases.

Kaf ha'Chayim (14): She'alas Shmuel was unsure about one whose Tzitzis snapped on Yom Tov Sheni. The Torah permits to fix it! However, Kavod ha'Briyos permits transgressing passively! He ruled that he should try to lend his Talis to Ploni and borrow Ploni's Talis (then both are exempt). If not, he may keep it on.

Note: The Magen Avraham said that when Tzitzis are not available in the city, it is like Shabbos. Surely, the same applies here!