Take tots for a spin on the carousel at the LVIS Fair with an old-fashioned playland, auction, music and shopping (June 8). The Hamptons JazzFest Kick-off Party (Bay Street Theater, June 10) inaugurates its July-September season. Enjoy Sag Harbor’s Spring Arts & Crafts Fair (Marine Park, June 15-16); Guild Hall Clothesline Art Sale (June 22), and Palm Tree Music Festival (Francis Gabreski Airport, June 22). 95 Main Street, East Hampton, N.Y., 11937. 1 Long Wharf, Sag Harbor, N.Y., 11963. Marine Park Drive, Sag Harbor, N.Y., 11963. 158 Main Street, East Hampton, N.Y., 11937. Sheldon Way, Westhampton Beach, N.Y., 11978.