The Shulhan Aruch in Siman 308:7-11 discusses the Mukse status of utensils that generally would be discarded, for example, a broken needle. Generally, such items are Mukse on Shabbat, even though they may have some other minor function.
The Poskim discuss whether a perfectly useable vessel that was thrown away before Shabbat, for whatever reason, becomes Mukse. For example, if someone throws a nice shirt in the garbage before Shabbat, does it now become Mukse on Shabbat, if someone would want to take it out? The Tosafot rule that it does not become Mukse; since his action was a deviation from the norm, it is overridden by the common sense of the general population. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) discusses the case of someone who threw out a perfectly good sitting mat and concurs with this ruling.
SUMMARY
A perfectly good item that was thrown away before Shabbat, does not become Mukse.
