Would it be legally and socially possible to have no surname in Regency-era England?

This is a really minor thing in Wuthering Heights I've always wondered about. There are many cases in later time periods of people who don't have a full first name or surname being made to adopt one for things like military papers, housing deeds, or passports, but was this also how it worked during the Regency era? Would it actually be legally and socially feasible for Heathcliff to only have one name?