I think referendums and plebiscites on specific issues are always a possibility even without the whole system being a direct democracy. I've always wondered why other countries have used them but the US never has.
Historical reasons, United States has always been uncomfortable with Democracy despite all our cheerleading around it. US Founders had a fear of populist sentiment leading to "rash" decisions and thus United States system has always been setup as bulwark against it. Entire Senate elections and rules serve as bulwark. Electoral College is another bulwark. Constitution Amendment process another bulwark. Our acceptance of leaving many issues up to the court like Abortion, Gay Marriage and Immigration. List goes on and on.
The US federal government does not, but they are rather common in US states. The US federal government was structured with its direct constituents as states, not citizens, and while that has evolved a bit over the years, it still is reflected strongly in its structures (Constitutional amendments must be ratified by a kind of referendum, but it is a referendum of state legislatures, not citizens.)
I don't know of cases where the US federal government uses them, but it isn't that uncommon for state or local governments to have them. Although the laws around them differ between jurisdictions.
