Gemara

(Mishnah): The four Tzitzis (sets of strings on each of the four corners of a garment) are Me'akev each other, because they are all one Mitzvah;

R. Yishmael says, each of the four is a Mitzvah by itself.

37b (Ravina): They argue whether or not Rav Huna's law applies (if there are threads on some but not all four corners);

(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).

(Rav Yehudah): The Halachah follows R. Yishmael.

The Halachah does not follow him.

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!

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)!

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 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.

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): Some of my Rebbeyim would remove their Talis before going into Reshus ha'Rabim on Shabbos. I say that this is a stringency without reason. If they are concerned lest strings snapped and Kosher Tzitzis do not remain, how were they allowed to bless on it? There is a great punishment for a Berachah l'Vatalah! If they checked before blessing, they can check before going to Reshus ha'Rabim! The custom is not to check before blessing. We leave it in its Chezkas Kashrus, until we know that it became Pasul. Tzitzis are not easily disqualified. R. Tam says that if one or two strings on a corner snapped, it is Kosher. If all of them snapped and Kedei Anivah (enough to wrap a string around the others) remained, it is Kosher. The custom is to fix strings before they become Pasul. (One who normally does not walk four Amos without Tzitzis should rely on the Chazakah and wear it in Reshus ha'Rabim on Shabbos.) If he wants to be stringent, he can check it before going out. Some say that all of our Talesim do not have proper Tzitzis, therefore, one may not go to Reshus ha'Rabim with them. This is wrong. We bless on them! The Torah said to put two strings of Lavan (i.e. not Techeiles) and two of Techeiles on each corner. A tradition from Moshe from Sinai teaches that neither is Me'akev the other. We put another two of Lavan to commemorate Techeiles. Even if these people were right, and our Tzitzis were improper, one may go in Reshus ha'Rabim with them. We permit (Shabbos 139b) to go in Reshus ha'Rabim wrapped in a Kilah. Pasul Tzitzis are different. They are esteemed, so they are not Batul to the garment. The Ri explains that Tzitzis are not Batel because he intends to fix the snapped ones and join them to the intact ones. Straps are Batel to a Kilah. Similarly, one may go in Reshus ha'Rabim with straps tied to a sash, even if he does not tie with them, for they are Batel to the sash. Similarly, our Tzitzis are Batel to the garment, even if one thinks that they (all our Tzitzis nowadays) are improper.

Teshuvas ha'Rashba (attributed to Ramban, 209): One who walks in Reshus ha'Rabim with Pasul Tzitzis is liable. Rashi explains that this was in the days when they had Techeiles. It was esteemed, and it was not Batel to the garment. Our Tzitzis are not esteemed, so they are Batel to the garment. R. Chananel explains that even our Tzitzis are esteemed, and they are not Batel to the garment. All permit going in Reshus ha'Rabim with a Talis Katan with Kosher Tzitzis. Maharam was stringent not to, lest a string break.

Nimukei Yosef (on Hilchos Tzitzis of Rif 14a DH Zocheh, citing the Ritva): Some are concerned nowadays on Shabbos to wear a Talis, for we do not have full Ituf (wrapping). It is under the outer garment. It looks like a load, i.e. a double (extra) garment. This is not a concern. It is a proper garment. Maharam was stringent lest the strings snapped and became Pasul.

Poskim

Shulchan Aruch (OC 13:1): The four Tzitzis are Me'akev each other. If all four do not have Kosher Tzitzis, one who enters Reshus ha'Rabim on Shabbos while wearing it is Chayav Chatas.

Beis Yosef (DH Arba): Rashi explains that all agree to Rav Huna's law.

Taz (1): If all the Tzitzis snapped and Kedei Anivah does not remain, surely he will make new Tzitzis, so they are Batel to the garment and one may wear it in Reshus ha'Rabim.

Mishnah Berurah (3): If any strings are Kosher, he intends for them (to use them), and they are not Batel.

Kaf ha'Chayim (1): The Levush permits if nothing remains. This connotes that if even less than Kedei Anivah remains, it is forbidden. Eliyahu Rabah says that even the Taz permits only regarding Shabbos, but it is forbidden due to Bitul Mitzvas Tzitzis, unless Kavod ha'Briyos applies. Piskei Olam questions the Taz. Since he will later cut off the strings (because he wants to put new strings in), they are not Batel! Therefore, one should be stringent like the Levush.

Mishnah Berurah (1): All of them are one Mitzvah. If even one of the (corners with) Tzitzis is missing, it is as if he did not put Tzitzis, and he was totally Mevatel the Aseh of Tzitzis.

Mishnah Berurah (2): The same applies to other Pesulim of Tzitzis, e.g. the garment is mostly closed, the Tzitzis are not the correct distance from the corner, or the garment is too small. One may not enter even a Karmelis with such a garment.

Maharam Mintz (110): If a Talis Katan is smaller than the Shi'ur to be obligated in Tzitzis, the Tzitzis do not fulfill a Mitzvah. They are like shoe straps. (They are Noy of the garment, and are Batel to it.) However, perhaps they are different. The garment can be fixed (enlarged) and it will be obligated in Tzitzis, or the Tzitzis can be removed and put on a garment that is obligated.

Magen Avraham (2): Maharam Mintz (110) was unsure if it is permitted if the garment is smaller than the Shi'ur that is obligated.

Mishnah Berurah (9): If on Shabbos one was in Reshus ha'Rabim and found that the Tzitzis on his garment are Pasul, he must remove it even if he will be left naked. Kavod ha'Briyos does not override a Torah Isur like this. Some say that nowadays we do not have Reshus ha'Rabim mid'Oraisa. If the windings are intact, and it is Pasul only in what hangs straight after the last knot, one can join the opinion of the Ri, who is Machshir. He need not remove it immediately, but he must hurry home.

Kaf ha'Chayim (8): Sha'agas Aryeh (58) says that in a Reshus ha'Rabim he must remove it, for it is an Isur Torah. The same applies if he found that he is wearing Sha'atnez. Aruch ha'Shulchan (3) says that if one sees that Ploni's Tzitzis snapped in Reshus ha'Rabim, Ploni is Shogeg, so he need not tell him.

Shulchan Aruch (2): If a Talis Gadol or Talis Katan has Kosher Tzitzis, one may wear it in a in Reshus ha'Rabim.

Magen Avraham (1): This is even at night, when the Mitzvah does not apply, for the Tzitzis are Noy for the garment.

Mishnah Berurah (4): One may wear even garments of other materials (than wool and linen), which are obligated only mid'Rabanan. The Tzitzis are Noy for the garment, since it is a definite obligation. It is unlike a garment that is half open. That is obligated only due to Safek, therefore, one may not wear it in Reshus ha'Rabim (OC 10:7). This is unlike Shev Yakov, who forbids garments of other materials.

Rema: We leave the garment on its Chazakah that the Tzitzis are Kosher. One need not check it before going outside on Shabbos.

Mishnah Berurah (7): Since he checked it in the morning before blessing on it, and it was Kosher, it is unlikely that it became Pasul in a short time, even if he took it off in between (Magen Avraham). This shows that also on Shabbos one must check. People are not careful about this.

Bi'ur Halachah (DH Kodem): Perhaps people are lenient about checking it on Shabbos, for even if it will be found to be Pasul, the Rema (Sa'if 3) permits wearing it in the Beis ha'Keneses. This is like the Magen Avraham, who says that one checks lest he wear a garment without Tzitzis. However, the Taz says that the concern is for Berachah l'Vatalah. If so, he should check! Also the Magen Avraham is lenient only in the Beis ha'Keneses, but not to wear it in his house! Also, if it is Pasul, he should know not to wear it in a Karmelis or Reshus ha'Rabim! It is proper to check it on Erev Shabbos when putting it away, and then he need not check it on Shabbos.