According to the Shulchan Aruch (9:1), the only way to fulfill the Misva of Sisit from the Torah is with a Tallit made of wool. While the word wool is not mentioned explicitly in the Pesukim regarding Sisit, Chazal derived from other Misvot that the word "Begged" in the Torah (garment) denotes only a sheep’s wool. Any other fabric would only be obligated in Sisit M’drabanan, from a Rabbinic Misva.
However, The Rema argues and rules that almost all fabrics, including cotton, fulfil a Torah obligation. Thus, there is a divergence of opinions between the Sepharadim and the Ashkenazim. Nevertheless, Rav Mosh Feinstein (Russia-New York, 1895-1986), in a famous Teshuva in Iggerot Moshe (Vol. 1 and 3), rules that even Ashkenazim should be strict and fulfill the Misva using wool. He cites a Gemara in Menachot (41) which states that Sisit is, in effect, a voluntary Misva. That is, one is not obligated to go out of his way to wear a four- cornered garment in order to fulfil the Misva of Sisit. However, the Gemara continues and says that in an "Idna D’Ritcha"-a time of Divine anger, Hashem even punishes for not fulfilling a voluntary Misva. Rav Moshe reasons that since we don’t know if this is a time of wrath, it is best to fulfil the misva in the optimal way according to all the opinions, using wool.
Hacham Ovadia points out that if Rav Moshe recommended that Ashkenazim, who generally follow the lenient opinion of the Rema, wear wool, how much more so Sepharadim should be scrupulous to do so. Moreover, wearing a wool Tallit also enables one to avoid the dispute amongst the Rishonim (early authorities) as to whether a cotton Tallit requires strings made of wool or cotton. While most authorities agree that wool strings are required, the MaHaram M’Rotenberg held that a cotton Tallit requires cotton strings. Therefore, all of the Poskim say that one should try to use a wool Tallit, especially today that there is a light-weight wool that is not too hot in the summer. Hacham Bension writes in Or L’Sion that if someone feels that any wool is too hot for him, he should at least wear it in the morning and then change into cotton Sisit for the rest of the day.
The Vilna Gaon(Rav Eliyahu of Vilna, 1720-1797) had an unusual position on this issue. He was careful to only wear a cotton Tallit. This is surprising, since he was so scrupulous at following all of the most stringent positions. The Hazon Ish and the Steipler Rebbe also followed his path. Several explanations are offered as to why he did so. One possibility is that the Gaon would wear a linen undershirt, and he was concerned that wearing a wool Tallit over it might constitute a problem of Shatnez according to Rabbenu Yonah. The weakness with this theory is that it does not explain why he had to wear a linen undershirt. Others say that since the Vilna Gaon would wear his woolen Tallit Gadol all day long with his Tefilin, he was not concerned regarding the fabric of the Tallit Katan. The third theory is that he wanted to demonstrate to the common people, who thought that only wool was obligated in Sisit, that even cotton is obligated in the Misva.
It goes without saying that everyone should wear a Tallit Katan. One who does not is taking a great risk. As we live in dangerous times, which could qualify as "Idna D’Ritcha," the Tallit Katan serves as a spiritual protection, no less important than the other security measures we take to protect ourselves. Everyone should make sure they follow Shulhan Aruch and have a wool Tallit Katan with wool strings. If someone refuses and gives an "ultimatum"-either cotton or nothing, of course, cotton is preferable to nothing. However, just as a person tries to do other things in his life in the best fashion, so too Misvot should be performed in the best possible way.
Everyone should be sure to wear a Tallit Katan made of wool.