Question: I was wearing a new suit on Shabbat and I noticed a random thread hanging from it. Is it permissible to remove this thread on Shabbat?
Answer: The Gemara (Shabbat 75b) states that one who removes the heads of threads hanging from a new garment on Shabbat is liable for the biblical forbidden work of “the final hammer’s blow,” i.e. putting the finishing touches on a vessel or garment. Thus, since tailors and clothing manufacturers usually remove such hanging threads from the garment after it is sewn and they had probably just forgotten to do so, it would therefore be forbidden for anyone to do so.
It would therefore be forbidden to remove this thread on Shabbat. Only if this was done unintentionally while merely handling the garment would this not have constituted a Torah prohibition.
An Additional Question: May one unclog a sink on Shabbat?
Answer: The Acharonim discuss the matter of whether or not it is permissible to unclog a sink or toilet using a plunger on Shabbat lengthily. Hagaon Harav Shlomo Zalman Auerbach zt”l rules leniently on this matter, however, Maran Rabbeinu Ovadia Yosef zt”l rules stringently (and he repeated this discussion in his Chazon Ovadia- Shabbat, Volume 5, page 331) and writes that this constitutes a Torah prohibition of repairing a vessel on Shabbat. He proceeds to support this with many proofs.
Hagaon Harav Menashe Klein zt”l writes in his Responsa Mishneh Halachot that when he visited Israel, he visited the home of Hagaon Harav Auerbach and discussed this matter with him. Harav Auerbach told him that he ruled leniently on this matter but that his dear friend, Harav Ovadia Yosef, ruled that this constitutes an actual Torah prohibition, and thus, even Harav Auerbach zt”l was concerned about a possible prohibition in this regard.
Nevertheless, Maran zt”l writes that if the drain is not completely clogged and the water is going down, albeit very slowly, in this case, it will be permissible to unclog the sink in a regular, unprofessional manner on Shabbat (such as by using a plunger).
Summary: It is forbidden to remove an extra hanging thread from a new garment on Shabbat. It is forbidden to unclog a clogged drain on Shabbat. (In cases of great need, one may instruct a non-Jew to unclog such a drain or toilet on Shabbat.)