Nations usually act on a single motivating factor: self-interest.
Given that we're asking this question on a US forum we can take it as granted that 60% of the readers couldn't find either country on a map and that 90% have zero knowledge of their political and historical position. So about 90% will be ill-equipped to understand where Singapore's and South-Korea's self-interests might lie. But now that the question is asked, we can remedy that.
South Korea, needs the US to help defend themselves against neighbours who would be prepared to wage a full-scale war against them (North Korea). The US are pretty much the only ally of note and value they have, and they know it.
Singapore is surrounded by neighbours that completely dwarf them (Malaysia, Indonesia) only 50 years ago encompassed them (Malaysia), have an Islamic majority (Malaysia) or a virulent Islamic minority (Indonesia) and are debating whether to become a fully Islamic state (Malaysia).
Both countries have brought about an economic boom and depend on security (i.e. the absence of shooting wars), good trade relations with the West, open sea lanes and suchlike.
In both cases a critical part of their national security is having accurate information on what their neighbours are really up to. And in both cases the only serious partner is the US. As a stabilizing factor, a main ally, or a party with whom to trade information that they themselves cannot collect (like e.g. satellite coverage, ocean reconnaissance, comprehensive traffic monitoring etc. etc.).
For countries like that, helping the US eavesdrop on message traffic makes an uncommon lot of sense and is a small price to pay.
Whilst Snowden's relevations may have a beneficial effect on US *domestic* intelligence oversight, having such data-collection arrangements splattered on the front page are detrimental to the collective national security of the US, Singapore, and Korea.
Turn it any way you want, knowing what people are up to gives you a head start in dealing with them, and the US have been a stabilizing factor in Asia for 60 years or so. Eroding this data-collection capability is the price we pay for openness. I'm not certain if the price is too steep, all I'm saying is that it's a very real price we pay. Even if not everybody realises it or wants to hear about it.