Offhand I can think of Iceland, Java (Indonesia), and parts of Afghanistan.
In Iceland, 90% of people's names include a first name and a patronymic (reference to dather's first name), but not a "last" name per se. So Sigurd Thorsson is "Sigurd the son of Thor", but his dad might be Thor Svensson (Thor the son of Sven). Sigurd's sister is Helga Thorsdottir (Helga the daughter of Thor). But since these 2nd names are entirely dependent on one's own father and don't usually repeat, they're not "surnames".
In Java, many people have either a name structure similar to the Icelanders (a "son of" or "daughter of" referent after the "real" name)or simply have a single name. Presidents Sukarno and Suharto of Indonesia were Javanese and their full names were simply "Sukarno" and "Suharto".
Some percentage of people in Afghanistan have only a single name; my guess is they're too poor to afford another. : )
As to whether it's legally possible to have only a single name in the US, I can't say. But since we're all SS numbers at this point anyway, I can't see what harm it could do.