The sanctity of tzitzit strings
Question
Answer
Shalom u’vrachah.
According to the basic halachah, tzitzit strings do not have inherent sanctity; however, one must be careful not to treat them disrespectfully. Therefore, they should be wrapped in a bag and thrown into the garbage.
Indeed, some are stringent and place them in genizah, as an added honor for the mitzvah. One who is stringent in this matter shall be blessed.
Source
Shulchan Aruch (Orach Chaim, siman 21, se’if 1):
Tzitzit strings that have torn may be thrown into the garbage, because this is a mitzvah whose object itself has no inherent sanctity. However, as long as they are attached to the tallit, it is forbidden to use them—for example, to tie something with them or the like—because of disrespect for the mitzvah.
Rema: And there are those who say that even after they have torn, one should not treat them disrespectfully by throwing them in a disgraceful place; however, they do not require genizah (Kol Bo, Laws of Shabbat). And there are those who are meticulous to place them in genizah; and one who is stringent and meticulous in mitzvot shall be blessed.