The best I could find in the books is that the Frost brand (171, DMG) Art depicts the sword as a scimitar (the Frost brand is a magic weapon that can be any sword). And the description of the bladesinger cat style:
Cat. Styles that employ a sword belong to this family. (...) Red tiger, a style just three centuries old, has its bladesingers using the scimitar in a whirling dance of defense (142, Sword of the coast adventure's guide)
unfortunately they look more like fluff. And I couldn't find anything specific about the rapier.
But maybe we can use some history to classify the rapier as a sword. The word rapier, according to wikipedia:
The word "rapier" is a German word to describe what was considered to be a foreign weapon
The rapier first name was the spanish word "Espada ropera" or "dress sword". The people of that time also called the dress sword just "Espada" (Which translates to "Sword").
A sword consists of a long blade attached to a hilt. The blade
can be straight or curved (...) The thrusting swords such as the
rapier (...) (Wikipedia: Sword, Emphasis mine)
Non-European weapons called "sword" include single-edged weapons such
as the Middle Eastern scimitar (...) (Wikipedia: Sword, Emphasis
Rapier or Espada Ropera, is a loose term for a type of slender,
sharply pointed sword (...) The word rapier was not used by Italian,
Spanish, and French masters during the apogee of this weapon, the
terms spada, espada, and épée (or espée) being instead the norm
(generic words for "sword") (...) (Wikipedia: Rapier,