They have family naming conventions — but not exactly surnames.
Both on-screen and in supplementary material and interviews, we get hints or outright information about some of their families' naming patterns. Sadly, some families are more mysterious than others, but it seems reasonable that the ones we don't know about yet do also have some kind of common moniker.
At first this family seemed to come closest to having an actual surname. Pinkie Pie mentions her "Granny Pie" in the Laughter Song, and her sisters are Marble Pie and Limestone Pie (an interview with Faust revealed that Pinkie's sisters might have been named Mince Meat Pie and Chicken Pot Pie).
However, her parents are named after stones (Cloudy Quartz and Igneous Rock). Maybe they've got an every-other-generation thing going on? I got nothin'.
This is the most obvious, and while "Apple" is definitively not a surname, it is clearly their family name and the common connection between their individual names. "Apple" is more like a clan name than a surname.
Rainbow Dash we aren't sure.
RD's past is shrouded in mystery. In flashbacks (particularly "Games Ponies Play") we've seen her associated with the adult pegasi Rainbowshine and Rainbow Blaze, and other media like collectible cards reinforce that relationship, but these ponies aren't named in the show itself. Presumably their family would share a "rainbow" naming convention.
Rarity we can't figure out.
Her mother seems to be based on two previous-generation ponies who have "Wish" in their names,but a mobile game calls her "Betty Bouffant" and Rarity's father "Magnum." And then there's Sweetie
Bot Belle... this family doesn't have any obvious naming conceit.
Fluttershy we don't know.
Like, anything. At all. The girl's family is a closed book.
Twilight Sparkle's family is named for kinds of radiance.
Obviously we've got Sparkle, Twilight and Armor, Shining. Their mother was named Twilight Velvet in a mobile game and in mystery pack toys, while their father is nameless but seems partly based on a G1 German toy called Nachlicht (Nightlight).
[For the record, while Western names are often of the "given + sur" construction, many other cultures work differently. I personally live in a culture where extended relations have a "familian" name in common, regardless of what surnames they each have: familian babui (the boar family) is a broad description of clan-like kinship. The Apple family naming convention reminds me of this.]