In the times of Ezra/Nechemia, when they found the people weak in observing Shabbos, they placed a temporary ban from moving even כלים שמלאכתם להיתר. The gemara finds certain scenarios in the Mishna when it appears that even regular כלים are forbidden to use on Shabbos and the Gemara suggests the Mishanyos were a reflection on these temporary restrictions.

Even after Nechemias laws were eased and כלים שמלאכתם להיתר can be moved on Shabbos, we still only move the כלים only if there is an actual need to use them. For example, when flaying the Korban Pesach on Shabbos, they wouldn’t use staffs to suspend the carcass bc they could just use each other’s arms and hang the animal over them.

There’s one tanna R’ nechemia (not connected to nechemia) who holds that even כלים שמלאכתם להיתר cannot be moved unless it’s for their designated use. Acc to this view, a regular kli cannot be moved to use its place. This is so extreme that the Gemara wonders how one can move dirty pots and plates away from the table on Shabbos! The Gemara explains there’s a special leniency in muktzeh called גרף של רעי- repulsive muktzeh can be moved.

A כלי שמלאכתו לאיסור can be moved לצורך גופו. There’s a dispute if it can be moved לצורך מקומו but the prevailing opinion is that it may be moved. However, this is only for כלי. But a piece of wood is totally muktzeh on Shabbos and cannot be moved at all bc it’s not a כלי. On Yom tov, wood is prepared for firewood (permissible on Yom tov) and therefore not muktzeh. However, Beis Shammai makes a decree to treat the wood as muktzeh on Yom tov to protect the laws of Shabbos.

When a kli breaks on Shabbos, there a dispute. Tanna Kamma says as long as the shard is suitable for some usage, it’s not muktzeh. But R Yehuda says it must be suitable for the same general usage the original kli was made for (like fragment of glass kli that can hold some oil). If it’s not suitable for the same usage, it’s called נולד- a new entity and a form of muktzeh.

Leftover bricks from construction could become not muktzeh bc now they will be used or sitting. But if the owner stacks them up it means they’re designated for future construction so they’re muktzeh.

The shards of a broken kli are not muktzeh acc to Tanna Kamma bc they’re fit for some random usage. (Like covering a kli) The Gemara has a dispute whether this is true even if the shard is not in a location that has many keilim to cover (like in a street).

If someone throws a kli into the garbage it doesn’t become muktzeh bc it’s perfectly usable. But if they throw a broken shard in the garbage before Shabbos then it becomes muktzeh bc it has been thrown out.