Magirhy,
Here is my answer to your questions—which are clear for professional working in the field" ;-)
==== "Search the Web" vs "Search pages from Singapore" ====
You said that your pages appear in the global index—Search the Web—at not in the local index—Search pages from Singapore.
Here is the reason why.
But before, you simply need to step back and quietly listen to what Google is telling you—it doesn't qualify them for local search. My guess is that you have a .com domain, maybe hosted outside of SG—all of which doesn't give a hint to Google about the audience you target.
Google—among other search engines (SE)—says that the top-level domain (TLD)—e.g., .com.sg—is sufficient to help relate the site to a country—and in most cases, a language.
And lucky you, Google allows you to explicitly target a country for your site—regardless of its TLD—within Google Webmaster Tools (GWMT) (see link below). In which case, Google says the TLD doesn't matter. In fact it helps Google localize a country-agnostic site—i.e., .com, .org, .net, .info.
Aside from explicit localization in GWMT or from the TLD of your choice, Google says that the IP address of your site—something link 66.201.244.33—is also part of the localization equation they use. So it's better to host it as close as possible to Singapore—ideally locally or at least in Asia.
In a nutshell, you don’t need to register a .com.sg but merely:
1. geotarget your site in GWMT (see link at the bottom on how to link a site to a country); and
2. create a profile in Google Local Business Center (GLBC) associated with this domain (see link at the bottom on how to create a GLBC profile).
==== Provided description vs. Google’s own page description ====
Google uses the description provided in the page metatag ONLY when the keyword combination searched is found within it—otherwise, Google comes up with its own description built from occurrences of words within the page content.
Is there a way to ensure Google only shows the page with the carefully crafted page description?
—NO.
As I said, Google will come up with its own description. So you have to make sure to embed in your description all the keywords you deem important and relevant for your site in order to maximize the appearance of the carefully crafted description in search engine results pages (SERP).
Note: Geotargeting a site doesn't exclude it from the global catalog and it will still appear in international search; the opposite is not true though—an untargeted site is disadvantaged compared to geotargeted and country-TLD'ed sites when searches are performed against the local index.
—Pascal