Steve,
I've the same question, few days ago I tried to use the singleton in ABAP. In Java programming is possible to use the same reference in two sessions or programs, sharing attributes, methods and all data, but I could not do in ABAP.
In my test I created a program with one global class using the singleton pattern, so I expected that when I run my program and see the reference returned after the get_instance method it should be equal to the same program run in another session, but the ABAP will create a new reference and instantiate again.
So I deduced that the only way to share it between sessions in ABAP is using the ABAP Shared Memory Objects.
I can be wrong, but I think that ABAP use a thread by user session (Each window) and we can't change it.
Best regards.