A twist tie is a metal wire that is encased in a thin strip of paper or plastic and is used to tie the openings of bags, such as garbage bags or bread bags. A twist tie is used by wrapping it around the item to be fastened, then twisting the ends together (thus the name).
Are these allowed to be used on Shabbos? Or is it the same as tying a knot, which is prohibited? Would it matter how many twists are made?
I have been taught that there is no problem if the tie is wound round but not twisted. This works fine. That twisting is a problem is discussed here. Winding round as a solution is mentioned in Pathway to Prayer, Sefardic, Shabbos by Birnbaum see para 12d.
It's a machlokes haposkim if one can use them for 24-hour knots. R'Ribiat discussed this issue in "The 39 Melachos" (p.799-800). He says one can use them, but it is praiseworthy to avoid it if they will remain tied for more than 24 hours. The reason to permit it is since its a mode of use, it may not be a melacha. Also, it stays in place because of the wire's stifness, unlike a regular knot.
However, I think this case may be one of the Chumrei R'Willig and the Kulei R'Ribiat.
Update: See Torahmusings
Rav Nachum Tzvi Kornmehl in his shu"t Tiferes Tzvi is matir. I do not remember exactly which chelek or siman and I could not find a link online. I do remember that he argues with the comparison to the Rambam brought in @Avraham Yitzchok's link, and compares it to an alternate Rambam instead.
I would assume that separating attached ties into its intended post-consumer usage would be forbidden (probably d'rabbannan as it is far from being a true keli).
Without any sources or additional reasoning I've heard the following a few times said over by HaRav Shammai Gross Shlit"a (one of the Gedolei HaPoskim in Eretz Yisrael who had a kesher (no pun intendend) with R' Eliyashuv.) One time when they asked R' Eliyashuv if it's muter to use the twisty ties on Shabbos he said it was ossur. Another time they asked him he said it was muter. They went back and asked then how could he pasken one time ossur and another time muter. He explained that when asked the time that he said it was ossur the person brought a twisty tie and demonstrated how it was used and twisted it (or perhaps rather "wound" it) around the bag many times. The time when he said it was muter the person came and asked and demonstrated as well however only twisted it (or "wound it") once.
