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.