Siman 21
Presented here for study, not as a practical ruling. The Shulchan Aruch rules by Sephardi custom, the Rema's glosses by Ashkenazi custom, and the Kitzur follows Ashkenazi custom. For a practical question, ask your rabbi.
א חוּטֵי הַצִּיצִיּוֹת שֶׁנִּפְסְקוּ יָכוֹל לְזָרְקָן לָאַשְׁפָּה מִפְּנֵי שֶׁהוּא מִצְוָה שֶׁאֵין בְּגוּפָהּ קְדֻשָּׁה, אֲבָל כָּל זְמַן שֶׁהֵם קְבוּעִים בַּטַּלִּית אָסוּר לְהִשְׁתַּמֵּשׁ בָּהֶם, כְּגוֹן לִקְשֹׁר בָּהֶם שׁוּם דָּבָר, וְכַיּוֹצֵא בָּזֶה מִשּׁוּם בִּזּוּי מִצְוָה. וְיֵשׁ אוֹמְרִים דְּאַף לְאַחַר שֶׁנִּפְסְקוּ אֵין לִנְהֹג בָּהֶם מִנְהַג בִּזָּיוֹן לְזָרְקָן בְּמָקוֹם מְגֻנֶּה אֶלָּא שֶׁאֵינָן צְרִיכִין גְּנִיזָה (כָּל בּוֹ הִלְכוֹת שַׁבָּת) וְיֵשׁ מְדַקְדְּקִין לְגָנְזָן וְהַמַּחְמִיר וּמְדַקְדֵּק בְּמִצְוֹת תָּבוֹא עָלָיו בְּרָכָה (מַהֲרִי"ל הִלְכוֹת צִיצִית) וְע"ל סי' תרס"ד סָעִיף ח' ט'.
1 What To Do With ציצית That Detached and With An Old טלית. Containing 4 Se'ifim: ציצית strings that detached may be thrown into the garbage, because it is a מצוה that does not possess it's own holiness (i.e. ציצית are only a מצוה by virtue of their being attached to a four cornered garment). However, as long as they are attached to a טלית it is forbidden to use them [for another purpose], like to tie something [with them] or similar, because this is disrespectful to the מצוה [And there are those who say that even after they become detached, one should not treat them disrespectfully by throwing them into a dirty place, but they don't need to be ritually buried (Kol Bo - Laws of Shabbat). And there are those who are punctilious to ritually bury them, and for the one who is stringent and careful in the performance of מצות, a blessing shall come upon him) (Mahari"l - Laws of ציצית)(And see below - siman 664, se'if 8 and 9)].
ב טַלִּיתוֹת שֶׁל מִצְוָה שֶׁבָּלוּ, אָדָם בּוֹדֵל עַצְמוֹ מֵהֶם, וְאֵינוֹ מֻתָּר לְקַנֵּחַ עַצְמוֹ בָּהֶם, וְלֹא לְיַחֵד אוֹתָם לְתַשְׁמִישׁ הַמְגֻנֶּה, אֶלָּא זוֹרְקָן, וְהֵם כָּלִים.
2 A טלית used for the מצות [of ציצית] that wore out, a person should not use them. And one is not permitted to wipe himself with them, and not to designate them for any disgusting use, rather he can throw them away, and they will wear out [on their own].
ג מֻתָּר לִכָּנֵס בְּצִיצִית לְבֵית הַכִּסֵא: הַגָּה: וְכָל שֶׁכֵּן לִשְׁכַּב בָּהֶן דְּשָׁרֵי וְיֵשׁ שֶׁכָּתְבוּ שֶׁנָּהֲגוּ שֶׁלֹּא לִשְׁכַּב בְּטַלִּית שֶׁיֵּשׁ בּוֹ צִיצִית, גַּם שֶׁלֹּא לִתְּנוֹ לְכוֹבֵס גּוֹיָה לְכַבֵּס וְהַכֹּל שֶׁלֹּא יִהְיוּ מִצְוֹת בְּזוּיוֹת עָלָיו אַךְ נוֹהֲגִין לְהָקֵל לִשְׁכַּב בָּהֶם (כָּל בּוֹ).
3 It is permitted to enter a bathroom with ציצית. Rema: And certainly to sleep in them is permitted. And there are those that write that it is our custom to not sleep in a טלית that has ציצית on it, also one shouldn't give it to a non-Jewish cleaner to clean. And all these things are so that commandments shouldn't become disgraced because of it. However, the custom is to be lenient [with regards] to sleeping in them (Kol Bo).
ד יֵשׁ לִזָּהֵר כְּשֶׁאָדָם לוֹבֵשׁ טַלִּית שֶׁלֹּא יִגְרֹר צִיצִיּוֹתָיו:
4 There should be caution, when a man wears a טלית, that he shouldn't drag it's ציצית [on the ground].