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To be close from both the nightlife and the tourist sites, stay in the downtown area near the Sule Pagoda.
One of the poshest neighborhoods in Yangon, it is also where you'll find the trendiest nightclubs and bars.
They can be found in the brand new mall Myanmar Plaza (Fuse, Eclipse, Harry's), in Kabar Aye Road (Cask 81, The West) or in Kanbawza Street (Genki Kids, Porter House and a few more fine dining restaurants). This is not exactly a nightlife area, but you have several rooftop bars and cafés targeting tourists and locals. Vista and The Penthouse have a beautiful view on Shwedagon Pagoda. On the main road, you may encounter ladyboys looking for customers.
Make sure you have a solid travel insurance that covers your whole trip. You can easily get a quotation in 1 minute: How Much Does Travel Insurance Costs? Within the Inya Lake Hotel compound, there are 2 upmarket nightclubs (DJ's Bar and Inya Bar).
A few hundred meters away, in Kan Yeik Thar Street, there are some less expensive bars and restaurants targeting both expats and wealthy Burmese (Brave, The Vibe, Joe's Kitchen, Spy and Escape).
Burmese drive like crazy and they'd run over you just to avoid slowing down.
Crossing Yangon from Chinatown to Inya Lake costs about 4,000MMK.
Traffic jams are a big problem during the day (average speed: 10 kilometers/hour) but it gets better after 8pm.
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Located on the rooftop of Yangon's newest mall, Fuse is easily the trendiest nightclub in the city at the moment. → Crowd: 80% Asian guys (Chinese, Koreans, Japanese, Burmese), 20% Western tourists → Prostitution: 100% → Entrance fee: 5,000MMK with one beer → Open every day from 7PM to 12AM When I visited, it was really empty. In particular, they hold LGBT friendly events and Hip Hop nights.