There are pros and cons to being in a country with very strict rules, but one of the pros is Singapore's commitment to conservation. We benefited from three of those projects, and four if you count the zoo: Fort Cannery Park, the Botanical Gardens, and the Nature Reserve.
The Botanical Gardens are free to the public, responsible dog-owner friendly, and very close to the Embassy. You could easily spend your lunch hour de-stressing in the green.
The Singapore Zoo is a delight for adults and children. So many monkeys!!
There is a family of orangutans who roam free in the treetops above the sidewalks. But watch out if one of them is above your head!
For an additional fee you can stay for the 7 pm Night Safari, which is in an entirely separate park next to the zoo. Even though tours leave every ten minutes, I recommend getting in line at 6 pm. You'll wait about an hour, but if you arrive at 6:30p you'll be in line for much, much longer.
Bukit Timah Nature Reserve, also free to the public, is a mostly undisturbed section of primary rain forest, and heaven to green lovers from Manila! Surprisingly polite wild monkeys roam about, and if you look closely you will see a flying lemur, or a Colugo, trying to blend in with a tree's bark. If you are especially lucky you might see one take flight.
Singapore is Manila's polar opposite: it's quiet, exceptionally clean, and orderly. If you're looking for 3rd world adventure it's not for you, but it is a welcome break to Americans living in Manila. However, you'd best follow the rules.
In Manila most rules are a suggestion and open to various ways of "coping" with getting caught, but Singaporeans take their rules very seriously.