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I haven't seen anything in the docs about when/if to use `remember {}` or `mutableStateOf()` inside state holders... or whether the one surrounding the object itself is enough....?

If associate each composible with class then:
val x = remember { initialValueExpression } - like val field in class,
var x  by remember { mutableStateOf(initialValueExpression) } - like var field in class (changing x will also trigger recomposition if x is used below)

Probably this explanations will help
<https://medium.com/androiddevelopers/jetpack-compose-stability-explained-79c10db270c8|https://medium.com/androiddevelopers/jetpack-compose-stability-explained-79c10db270c8>

Very interesting writeup! So to sum up the answer: When using properties that the compiler might take as mutable in the state holder, use `mutableStateOf`...

And if the state holder class needs to be instantiated in the Screen composable, then `remember { mutableStateOf&lt;MyStateHolder&gt;(MyStateHolder()) }` needs to be used there even though inside it there might be some `mutableStateOf()`s for mutable properties?

About when to use `remember` or not.
It’s simply a matter of if you’re inside a composable function or not. If you are, you have to think that whatever you’re calling can and will potentially be called many times a second. If you do `val x = Foo()` inside the context of a @Composable function, `Foo()` will be re-called on every recomposition and therefore if you’re saving anything in there, it will be overwritten.
If you are not inside a composable function then you just never need `remember`. Hence the examples where inside plain classes you can just do `mutableStateOf` directly.
That’s how I like to think about it at least.