And yet there are also places where Singaporean spotlessness 
  gives way to dirt. Places like Geylang, a red light district 
  well off the tourist map, and Orchard Towers, with its
  so-called "four floors of whores".

Actually, Geylang is hardly off the tourist map. It's quickly becoming more like the French Quarter in New Orleans--a rapidly gentrifying night spot with enough of the historical seedy element (in this case, many of the famous brothels with the hanging red lanterns that only a few years ago were far more numerous) to keep things authentic. Also a relatively affordable and increasingly popular place to live that's close to downtown.

FWIW, I know this because with all the new development it's a popular area for Airbnb rentals. Given it's reputation, I did some research as well as inquire with local family-in-law. As an outsider a very familiar picture emerges of a changing historic neighborhood. But opinions from Singaporeans might be far more mixed.