Gemara

26b (Tana d'Vei R. Yishmael - Beraisa): The Torah often says 'Beged' without specifying. In one place, it specified 'Tzemer u'Pishtim.' In all places, it refers to wool or linen.

(Rava): He teaches that garments of other materials that are three Tefachim by three Tefachim are not Mekabel Tum'ah.

27a (Abaye): The following Tana d'Vei R. Yishmael argues with the previous Tana d'Vei R. Yishmael:

i. (Tana d'Vei R. Yishmael - Beraisa) Suggestion: Perhaps "Veged" teaches that only garments of (sheep's) wool and linen receive Tum'as Sheratzim!

i. Rejection: "Oh Veged" includes also garments of camel's or rabbit's wool, goat's hair, silk, Kalach (floss silk) and frayed silk.

(Resolution #1: Rava): They do not argue. They agree that three by three fingers of other garments do not receive Tum'ah, but three by three Tefachim do!

(Resolution #2 - Rav Papa): The first Tana d'Vei R. Yishmael taught about Sha'atnez.

Rejection: Why would he have to teach that? It explicitly says "Lo Silbash Sha'atnez Tzemer u'Fishtim Yachdav"!

(Resolution #3 - Rav Nachman bar Yitzchak): The first Tana d'Vei R. Yishmael taught about Tzitzis.

Question: Why did he need to teach that? It is obvious!

i. Right after "Lo Silbash Sha'atnez Tzemer u'Fishtim Yachdav" it says "Gedilim Ta'aseh Lecha"!

Answer: One might have thought to learn like Rava;

i. Contradiction (Rava): "Ha'Kanaf" teaches that the Tzitzis (strings) should be the same material as the corner. (This implies that Tzitzis applies to garments of any material);

However, Tzitzis is written adjacent to "Tzemer u'Fishtim Yachdav"!

i. Answer (Rava): Strings of wool and linen are Kosher for any garment. Strings of any other material are Kosher only for garments of that material.

i. Tana d'Vei R. Yishmael teaches unlike Rava, that Tzitzis applies only to wool and linen.

(R. Acha brei d'Rava): Tana d'Vei R. Yishmael includes other garments regarding Tum'ah because it says "Oh Veged."

Question: We should include other garments also for Tzitzis. It says "Asher Techaseh Bah"!

Answer (Rav Ashi): That comes to include the garment of a blind man.

Yevamos 4b: Tana d'Vei R. Yishmael learns that one may wear Sha'atnez to fulfill the Mitzvah of Tzitzis from the extra words "wool and linen".

5b: Chachamim learn [that Aseh Docheh Lo Sa'aseh] from the Semichus of the Mitzvah of Tzitzis to the Isur of Sha'atnez.

Menachos 38b (Rami bar Chama): 'Lavan is not Me'akev Techeles' teaches about a garment that is entirely Techeles. The Mitzvah is to put Techeles first, for this is like "ha'Kanaf";

If Lavan was put first, this is not Me'akev.

Objection (Rava): It does not depend on the color of the garment! (We always put Lavan first.)

39b (Rachbah citing Rav Yehudah): Wool strings exempt a linen garment, and linen strings exempt a woolen garment;

Wool and linen strings exempt all garments, even silk, Klach or Serikin (carded flax).

Rishonim

Rif (Hilchos Tzitzis 14a): The Halachah does not follow Rava. We hold like Tana d'Vei R. Yishmael, that Stam garments in the Torah are of wool and linen. Camel's or rabbit's wool, goats' hair, silk and Klach are exempt. Rachbah taught that wool and linen exempt everything. Rav Nachman disagrees. The Beraisa says that they are obligated, i.e. mid'Rabanan. Rav Yehudah holds like Rava. The Halachah does not follow them.

Rambam (Hilchos Tzitzis 3:2): A Talis of other materials, e.g. silk, cotton, camel's or rabbit's wool, goats' hair or similar things, is obligated only mid'Rabanan, so people will be careful about Tzitzis. Stam garments of the Torah are only of wool or linen.

Hagahos Maimoniyos (6): My Rebbi (Maharam) told me that wool and linen exempt their own and other species only when they are together. (I.e. when we have Techeiles, and there is a Heter for Sha'atnez in Tzitzis - Kaf ha'Chayim 9:5.) Therefore, the custom is to make silk Tzitzis for silk garments. We are not concerned for the color, even if the strings are unlike the corner, like it says in Menachos, that the color does not matter. Rashi says that strings exempt only a garment of the same color. This is wrong.

Beis Yosef (OC 9 DH vea'Ha): The Agur brings from Maharil that many make wool Tzitzis for a silk garment, but not of linen, for Semak says so. Maharam says that one must put silk Tzitzis. The Terumas ha'Deshen (44) said that one of the Gedolim said that the custom is unlike Maharam, rather, like the Rambam, who explicitly permits. The Poskim connote like this.

Rosh (Hilchos Tzitzis 18): R. Tam, and Rashi in a Teshuvah, rule like Rava against Rav Nachman and Tana d'Vei R. Yishmael. In Yevamos 4b, we said 'this teaches unlike Rava.' That explained Tana d'Vei R. Yishmael. It seems that the Halachah follows Rava (for he is Basra, i.e. chronologically last). Why did the Rif rule like Rav Nachman against a majority, i.e. Rav Yehudah, Rachbah and Rava? If it is because one Tana d'Vei R. Yishmael holds like Rav Nachman, Abaye taught that another Tana d'Vei R. Yishmael disagrees!

Poskim

Shulchan Aruch (OC 9:1): Mid'Oraisa, only garments of linen or sheep's wool are obligated in Tzitzis. Garments of other matters are obligated only mid'Rabanan.

Beis Yosef (DH veha'Rav): The Rif rules like Rav Nachman, and also the Rambam does. This is why the Rambam did not bring a verse to teach that strings like the corner exempt other materials, for it is an enactment of Chachamim. Tosfos, Rashi, the Mordechai, Semag and the Rosh rule like Rava. Even though the Rosh was astounded why the Rif rules like Rav Nachman against a majority, since the Rambam agrees with the Rif, we follow them.

Mishnah Berurah (3): Even though one must bless also on a Mitzvah mid'Rabanan, there are differences whether the Mitzvah is mid'Oraisa or mid'Rabanan. 1) It is more proper to make a garment with which one fulfills the Mitzvah mid'Oraisa. 2) If strings spun Lo Lishmah became mixed (with strings spun Lishmah), and he has no other strings, or he has a Safek whether the garment is obligated, if it is obligated only mid'Rabanan, we are lenient about a Safek. 3) If he is among a Tzibur and one of the Tzitziyos snapped, on a weekday, if it is only mid'Rabanan, Kevod ha'Briyos overrides a law mid'Rabanan, so he need not remove the garment immediately. Mishbetzos Zahav (2) gives other differences. (If one has Talesim of silk and wool, and he blesses on the silk, he must explicitly intend to exempt the wool. Stam is not enough, even if he intended to wear both. Some say that nowadays if one wore Sha'atnez in Tzitzis mid'Oraisa, he transgressed only mid'Rabanan, so mid'Oraisa he is still Kosher for testimony.)

Rema: Some say that they are all obligated mid'Oraisa, and this is the Halachah.

Taz (2): The Rema rules like the latter opinion, for most Acharonim hold like him.

Gra: Rava is Basra. Even though Tana d'Vei R. Yishmael holds like Rav Nachman, another Tana d'Vei R. Yishmael disagrees, and also Rabanan (in Yevamos) disagree. The first opinion holds that since in Shabbos, the Amora'im discuss Tana d'Vei R. Yishmael's opinion, and Rav Nachman bar Yitzchak says that it comes to include Tzitzis, the Halachah follows him.

Mishnah Berurah (5): One who fears Shamayim should be concerned for the first opinion and make a Talis of wool, so it is obligated mid'Oraisa according to all opinions. It is proper that the Talis Gadol and Talis Katan both be of wool.

Kaf ha'Chayim (5): One who fears Shamayim should make only cotton Tzitzis for a garment of cotton, and similarly for other materials, to avoid the argument.

Doleh u'Mashke (p.23, note 61): The Chazon Ish wore a cotton Talis [Katan], like the Gra (Ma'aseh Rav 15, in Sidur ha'Gra p.506), for there is no Mitzvah if he dislikes wearing the garment, and to publicize that this fulfills the Torah Mitzvah. Also Ha'Gaon R. C. Kanievsky Shlita wears a cotton Talis.

Note: Also Ashrei ha'Ish (1:2:23, citing ha'Gaon R. Y.S. Elyashiv Ztz"l) says that there is no Mitzvah if he dislikes wearing the garment. Perushim on Ma'aseh Rav say that the Gra also wore a woolen Talis the entire day, and wanted to fulfill also a Mitzvah mid'Rabanan. Alternatively, he wore a linen shirt, and he was concerned for the Ge'onim who forbid wearing linen and wool garments without anything in between.