The 4-year-old Oasis Collections offers modern residences throughout South America for long-term rental along with a concierge service, members-only clubhouses, and private drivers. Of its 600 apartments, houses, and villas in eight cities, these three properties are among the best—and most stylish. 631.731.1677, www.oasiscollections.com

Rio de Janeiro, Brazil
This minimalistic home on Joatinga beach, which is known for its world-class surfing, features five bedrooms, six bathrooms, floor-to-ceiling glass …