Replacing troublesome global variable - are singletons acceptable?

I've inherited an old piece of software which makes extensive use of Global variables, which is seeing the usual issues associated with them.

A lot of these Globals I've managed to remove through passing them in (via constructor or method) however one variable (ID) is so ingrained in the application that passing it between objects and methods will become unmaintainable.

To attempt to mitagate this I have replaced ID with a method that can defensively check / provide ID wherever needed, as long as this is a singleton (via IOC container) this will work. I am however very concerned this is going to turn into a anti-pattern.

My questions are:

• Should I attempt to inject my singleton removing the Global nature of ID (pretty unmaintainable)?

• Should I use the much rejected 'service' pattern with IOC meaning I reference it directly within methods?

• Is this actually an acceptable time to use singletons?

This is very much a proof-of-concept at this point but I'd much rather do things properly.