Mevlevi dervishes perform the whirling sema ceremony in the Hodjapasha Art & Culture Center near Sirkeci Station (map), just down the hill from Sultanahmet, every evening at 19:30 (7:30 pm).
Admission costs TL60 for adults. The dervish music and whirling event lasts about one hour.
The Hodjapasha Art & Culture Center, in a restored Ottoman building (formerly a Turkish bath), is quite an atmospheric setting worthy of the solemn rite; and it's easier to obtain tickets to performances at the Center than it is for other locations.
The Art & Culture Center also hosts performances of Turkish dance on Tuesday, Thursday, Saturday and Sunday at 21:00 (9 pm) and Saturday at 21:00 (9 pm) The one-hour performance costs TL70; complimentary refreshments are served during intermission. An exhibit in the foyer outlines the 10,000-year history of dance in Turkey. More...
The center is right around the corner from the Hocapaşa restaurants, a good place for a moderately-priced dinner before or after the performance.
Take the Eminönü—Kabataş Tram from Sultanahmet to the second stop (Sirkeci), then walk south up Ankara Caddesi (away from the Bosphorus) to tiny Hocapaşa Hamamı Sokak on the left (map). Walk along this street, which turns right, and the Art & Culture Center is on the left.