It's midnight on a Saturday night at the Marina Bay Sands resort near the sparkling Singapore River, and all the boutiques are shut. But past a cosmetic-surgery clinic and a Ferrari accessories store close by, a large crowd is getting increasingly agitated. Dozens of hopefuls are clamoring to get in to what is billed as the world's most expensive club, Pangaea.

Tight-fitting Herve Leger bandage dresses are practically a uniform here, often paired with Christian Louboutin heels and Chanel 2.55 bags, as women try to befriend...