Several vendors sell tourist kitsch sea shells, beads on the Careenage at the Southern end of the Constitution Bridge next to the Independence Arch.
There are numerous stores including Cave Shepherd, the Macy's of Barbados on Broad Street...especially for jewelry. Most of these specialize in duty free shopping for citizens of the UK, Canada, U.S. and others.
Swan Street
A pedestrian only mall, has stores selling cheap clothes.
The Number One Music Shop
At the corner of fairchild street and bay street near the careenage has a wonderful selection of soca, reggae, calypso and other caribbean music including local bajan bands. also a good place to pick up tickets for concerts.
See also discussion of same topic for Barbados.