GRRRRRRRRRRRRRR IT’s not illegal to chew gum!! It’s illegal to IMPORT for distribution! And in my 15 or so years under the ban, I’ve never heard of anyone arrested for possession of gum! Or importing, for that matter, seeing as it’s the duty of any person going to Malaysia to bring back a carton (I estimate around 100 packs or more) or two of gum!
fume simmer simmer simmer
Okay, but drugs are nono. Really nono. You can get hung for importing drugs. Please don’t do that, I don’t want another trainwreck.
Okay, real live and fresh Singaporean here. Currently in the UK. I think there are at least 5 other Singaporean lurkers around, and more have worked a bit there but no longer.
Anyway, highly reccommended are the Zoo and the Night safari. I guarantee you these two will take up your entire day.
Boat Quay and Clarke Quay (;)) are the expat drinking spots.
River cruise is pretty… well… meh, I guess. Sentosa is nice for the old colonial stuff, if you’re into that kind of thing. I highly reccommend the Asian Civilisations museum on Sentosa itself, it gives an insight into the Chinese, Malay and Indian culture in Singapore, back in the old days. Sentosa itself is… well… kind of like a theme park, with a monorail that takes you around the small island. As has been said, the Underwater World is pretty impressive… or at least, it used to be. Slightly more run down now, after a decade or so, but still nice.
The musical fountain is kind of cheesy, but hey, the tourists seem to like it. 
Orchard road is the main shopping district, and it’s rather… dense. If you’re into shopping, you can easily spend a whole day there.
If you want to see ethnic culture, you can try and visit Chinatown, Little India, or Bugis, the old Chinese, Indian and Malay “enclaves”, before the policy of racial integration. I’m not sure if it’s good to go by yourself, however, as you’ll probably miss a lot of things. Get a walking tour, if you can.
I would probably skip the other museums, really, it’s not that interesting. But then, I have access to the British museum now, and I find that a bit dry, so YMMV.
2 days really isn’t all that much time. 
Travel:
If you’re thinking of renting a car, DON’T. Public transport is fast and cheap, and clean and sparkly. A bus/subway ride might range from S$1-S$2, which would be around… UD$0.6-US$1.5? ish? Not too sure. The exchange rate was 1.6 ish the last time I checked. However, if you’re in a hurry and not too budgety (If your swank hotel gives you all the free stuff, I assume so), a taxi should only cost about S$10 per ride around town. Of course, it’s metered, but that’s a fair price for trips near downtown.
What else… hmmm… In Singapore, it’s said that there are only 2 things to do: eat and shop. And you should definitely EAT! You can get Thai, Malay, Indian, Chinese, and some uniquely Singaporean dishes for cheap, if you avoid the tourist traps. It’s somewhat difficult to tell you where to go, but if you ask the hotel staff, they should be able to give you some reccommendations.
Seafood is generally good, and one of the famous dishes of Singapore is the Chilli Crab, or its cousin te Black Pepper Crab. Most people like those. Then you have the Indian and Malay curries, and the standard Chinese fare. No Kung Pao Chicken, though.
One traditional Singaporean dish is “Chai Tow Kueh”, or “Carrot Cake”. Also, you have “Char Kueh Teow”, or “Fried Kueh Teow” (Kueh Teow is a kind of noodle. Flat and made of rice). Your hotel should either carry a reasonable facimile in their restaurant, or a nearby foodcourt should suffice, if your hotel is in a downtown area.
Food Courts are a cheap and easy way of trying out as many different kinds of local dishes as you can. Each dish in a food court is about S$3-4, and is enough to feed one person. Well, a Singaporean sized, person, at least, you may have to buy two.
Just go into any food court, sniff around, and look at the things on offer. To order, just point at the pictures on the signboard, if there’s no english (There usually is in the foodcourts in town). I’m not sure about what food courts are like in America, but generally in Singapore most people eat in food courts or hawker centers, which are the dirtier, noisier and smellier predecessor of food courts. Cheaper and better tasting most times, though.
Bring sunscreen. Too many pink whales walking around. 
I’m not sure how much that helps, but if you want more info on any specific thing I will try my best to help.