I suppose I’m just as qualified to address this as anypony else. I’ve looked into the subject for what should seem like obvious reasons. Either way it’s not a short answer, so it’s after the break.

A lot of ponies have what are known as surnames. However, not all cultures have surnames in the same place. Sparkle is Twilight’s surname, but Apple is Applejack’s and Apple Bloom’s. If I were to hazard a guess, I’d say that AJ’s personal name is actually Jacklyn and she just shortens and puts the words together to play off of the well known cider (or possibly in reference to the breakfast cereal). I don’t know if there are, but if there were members of her family with Apple as the last word in their name, that would be romanticized. To be honest, I’m not sure about Big Macintosh or Granny Smith’s name. I don’t know if Big or Granny are parts of their names, but I could be fairly certain that Apple is either legally present somewhere, or they have no surname. More on that toward the end.

Then you have ponies like Rarity, who possess a surname but don’t regularly use it for whatever reason, giving the appearance that they only have one name.

Now this is a bit of an aside, but when marrying the old tradition was for the male to take the surname of the female. But after the Masculism Movement some time ago, the mare taking the surname of the stallion is no longer frowned upon but still less common. And of course in same-sex marriages there is no standard.

Then you have ponies who really do only have one name, or two or more personal names but no surname. I wouldn’t presume to know for certain, but for this example we’ll use Fluttershy and Rainbow Dash. Typically the reasoning for this is the family unit’s proximity to, or rather distance from, society at the foal’s birth. This allows for certain cultures to name their foals in unique ways far different from society’s norm. The other possibility, which seems to be gaining a small amount of momentum, is the conscious rejection of society’s norms to create a name that is radical and different. Something that the pony can be easily identified by for the rest of their lives. After all, a foal does not legally have to possess the surname of the parents.

Finally, you have where I am. Unidentified orphans aren’t legally given surnames, even if adopted. The idea being that if our identity is ever discovered, we can take our surname without any attachment to a different one.

There’s only one other possibility I can think of at the moment. One of the most interesting things I’ve learned about human culture is that you’re permitted to legally change your name. This is not the case in pony society. Once registered, your personal name will be your personal name for the rest of your life. The only option you have to change any part of your name outside of matrimony, is that an unmarried pony can drop their surname once they are of legal age.

And of course, none of this touches upon middle names like Pinkie Pie’s “Diane”, which are typically not used even if legally possessed.

One final thing to note is a very unfortunate outdated law that’s proving very difficult to change. As of today, two ponies or two anything, both without a surname, cannot legally marry in Equestrian society. It’s sort of a loophole. There’s been efforts to overturn that for years, but various factors keep impeding it.