I'm late, I'm late, for a very important deadline. Got a story here I want to get submitted, multiple occurrences of objects in it are on plinths, used many times 'sat' and 'set' and 'upon it', BTW blimey, doesn't the dictionary prattle on when you look up 'set'! Anyways, please someone, can you search mind and any vessel of knowledge for alternatives, been doing the _________ thing so I can put the words in later. Tempted to break boundaries and use made up words like 'plinthed' and 'entabled' if I can't turn anything up. A fire and forget post here sorry—checks stopwatch. Will pop back later see if anyone's been charitable. Stop.
I think the trouble is that when something 'sits' it's passive. Why is it there? Who put it there? What's it used for? Will it be re-purposed for the story (used for something it's not intended for)? Thinking along these lines, you might find a way to head in the direction of personifying the object so that it broods or skulks (or maybe sulks). Even waiting or holding its position is more active that just sitting there. Hope this helps.
On display/displayed, on view, on exhibit/exhibited, anchored, mounted, seated, located, positioned. Hope something there helps.
I can do something with your suggestions. Personification opens doors, and mounted (< why wouldn't that word come to me ) will be located in the pieces. Thanks guys.