During the Regency Era, a lady would never go out in a carriage and be seen in public without wearing the proper dress.
This is a carriage costume from November, 1819, as illustrated in La Belle Assemblee (Image from the University of Washington digital library.) The pink pelisse was made of figured gros-de-Naples and trimmed with the fur of an American grey squirrel. Click here to view more carriage dresses.
Two ladies in a high perch phaeton. The owners of these sporty, open-air and lightning fast carriages actually drove the vehicle, as there was no place for a coachman. Phaeton seats were built high off the ground, the sides of the vehicle were open to the elements (a top could be pulled over as a screen from sun or rain), and the back wheels were larger than the front wheels.
However, these light, airy, well-sprung vehicles were prone to tipping over when turning around corners too fast, thus a driver had to be skilled in order to move at high speed. The phaeton, therefore, was extremely popular with the rakish set.
Read more about transportation during the Regency Era at:
Carriages Carriages and their parts Transports of Delight: How Jane Austen's Characters Got Around, Ed Ratcliffe, JASNA
Click here to read another post about transportation on this site.