FYI there is a difference between "single payer" and "socialized medicine". The government runs socialized medicine. Single payer systems would run almost identical to what you see at a doctor in the U.S. Currently your doctor's office submits a bill to an insurance company. With single payer the same clinic sends a bill to uncle sam. You would probably use a drivers license (or other I.D.) instead of submitting an insurance card.
The U.S. Army could be called a "socialized army". The soldiers work directly for the federal government. Citizens do not have to send money to a insurance company for national security insurance. The FBI, state department, and commerce department are also "socialized". People do not talk about the army as "our socialized army" because the alternative is dumb enough to be unworthy of serious discussion.
Socialized meds: United Kingdom, Finland, Spain, Isreal, Cuba...
Single payer: Canada, France...