Use prc. (rc should be reserved for user input). The Private Request Collection is designed to be your "request scope" of sorts inside the ColdBox framework and it's automatically available to all action methods, all views, and all layouts. If you have a lot of extra methods in your handlers that aren't proper actions, those should really be in services but that's sort of a another conversation.
> I had thought that - akin to a singelton - the variables scope would be specific to the individual instance of the handler
That sentence doesn't make any sense-- a singleton, by definition, is only created once for the entire application and that single instance is shared for the entire app. So as you have found, there is only one instance of a handler CFC created. The only place it is safe to store variables specific to a single request is in a method local or method arguments scope. And for what it's worth, the rc and prc structs are pass in the arguments scope which makes them safe.
Thanks!
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