You will find a good selection of recognisable brands of underwear and lingerie in shops on-Island.
There are a few shops in Grand Cayman which sell lovely sets of underwear and bras and there are experts in those shops who can properly measure and fit you for a bra. The nice thing about buying underwear and lingerie locally is that is suits the climate. They also sell lovely nighties and PJs.
There is a dedicated lingerie shop in Camana Bay called Silhouette, and there is Atlantic Department Store, which can be found in Paddington Place off Godfrey Nixon Way which has a wonderful selection of underwear and is especially set up for teens and the fitting of their very first bra. They also stock cool and light PJs and nighties which are perfect for this climate.