Good safe birding and ideal for a 2/3 day stopover. In addition to Sungei Buloh and Bukit Timah and the Botanic Gardens, also try the Chinese Gardens (which I recall adjoin the Japanese Gardens). Easily and pleasantly reached via the train (MRT) system. I've birded here 5/6 times during stopovers and still generally add a few species, though as someone said earlier, don't expect huge day lists. The first time in the Chinese Gardens back in mid February 1995, the place was crawling with Yellow Bitterns perched on the reeds around the ornamental ponds and almost within touching distance, though I've never been there again in March, nor since seen them in such profusion. I saw my first Yellow-rumped flycatcher here in October of 1999. A morning stroll through the woods on Bukit Timah in Feb 1995 produced a Black Baza and a very confiding Banded Woodpecker near the summit. Later in the autumn of 1999 Bukit Timah produced a spanking adult Siberian Blue Robin among other goodies, such as the impressive Red-crowned Barbet hanging around also near the summit. In 1995 I walked around the fringes of a large golf course and picked up 4/5 new Bulbul species and Common Goldenbacks were very easy to see there A boat trip out to the islands takes about 35 mins and on several such outings I saw the impressively big Great-billed Herons (in both 1995 and in 1999) on the shorelines of several island we passed; - I rate Singapore as one of the easier place to catch up with this one. There is a nice mix of Sunbirds, Woodpeckers, Kingfishers (Common, Stork-billed, and Collared), Blue-tailed Bee Eaters, Dollarbirds, and a few Parrots/Parakeets, plus the odd surprise generally around, esp. in autumn. If you are tight for time try the Botanic Gardens, Chinese Gardens as the most readily accessible. Sungei Bulot and it's wader feast deserves a good 1/2 day minimum, and it's further out, necessitating an early bus; (or go in a hire car). Good hunting!
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In October wader passage will be underway so another option would be to take a MRT train then a bus to the Sungei Buloh wetland reserve (it was 10 years ago so can't remember the exact transport details). We saw 1000s waders including Marsh and Terek Sandpipers, Lesser Sandplovers, Pacific Golden Plovers and Grey-rumped Tattlers plus a nice cross-section of other mangrove and wetland species.
What's wrong with birding in Greater Manchester :) You can always discuss local stuff you know.
I went to Singapore in 1997 so it was a while ago now - there's not much 'wild space' left as it's a very modern commercial city. We went to the Botanic gardens which had a good selection of the common stuff like Mynas and White-eyes. Then the only real place to go is up by the reservoir at Bukit Timah where there is a reserve with monkeys and a good view from the top of the hill.
Your best bet is to search the web for trip reports and look on one of the more international forums like www.birdforum.net or surfbirds
If you have a long stay, then you're better off heading up the Malay peninsula to somewhere like Frasers Hill or do what we did and flew across to Borneo