dude. are you kidding? singapore is a bastion of naturally low-carb cuisines. in fact, most non-western ethnic cuisines are paleo-friendly. rice is served on the side and easily avoided; even excluding noodle dishes you have a ton of options.
how do i eat, as a singaporean caveman?
for lunch i do nasi padang or roast pork, within easy walking distance of the office. for nasi padang, which features chicken and beef cooked in coconut curries, i can recomend the stall at the southeast corner of north bridge road and kandahar street. if you say you're eating muslim food then you're probably know what i'm talking about. indian curries follow a similar format.
for chinese roast pork (think crackling) try "gourmet corner" on phillip street. they have the fastest moving lines in the business.
most sit-down chinese restaurants, if you go a la carte, separate the meats from the starches. for instance,
i hope this isn't too alien to you -- but if you don't want to die like a white man, don't eat like a white man: white sugar, white flour, white rice.
for dinner, i usually do a steak (grass-fed) or just three or four eggs scrambled in butter. this requires nothing more complicated than a stainless steel frypan and spatula, which will pay for themselves in a week. if you absolutely refuse to cook, there are lots of restaurants which will be happy to serve you fish and steak. but after a few months on the paleo diet it doesn't bother me to eat once a day. "three meals a day" is cultural conditioning. sure, breakfast is the most important meal of the day ... if you ask the American Corn Growers Association. but now i eat when i'm hungry and i stop when i'm not. not when i'm full. there's a huge caloric difference between the two.
If you're with a large group, consider steamboat hot pot: yet another ethnic cuisine which is 100% paleo if you just hold the rice and noodles. en.wikipedia.org/wiki/Hot_pot
Along those lines, Korean barbecue is also mostly meat and 5% green vegetables. en.wikipedia.org/wiki/Korean_barbecue
the thing that helped me most was not thinking of food as entertainment or stress relief. for smokers, for alcoholics, cigarettes and booze are a way to not deal with something that's bothering them. carbohydrates can be a way to assuage emotional needs that aren't being met anywhere else.
to be fair, carbohydrates aren't always an addiction. food is an important way to feel a sense of belonging to a culture: some societies don't eat pigs. others don't eat dog. for westerners, drinking beer and eating wheat- and corn-based foods are an important cultural marker. add to that fifty years of conventional wisdom and it's really hard to change anyone's eating behaviour who isn't already an iconoclast in some way.
Elaboration in response to comments:
Grass-fed beef can be found at any Cold Storage. The glass counters usually hold US grainfed; look to the refrigerated shelves nearby for pre-packed NZ or Australian pastured. (Those may still be grain-finished.) All the usual cuts are available. I don't shop at NTUC so I don't know if they carry, but if you can't find what you want at Cold Storage then go to either Jason's or the 360 place at Ion. Those are Cold Storage's deluxe groceries.
If you want to go artisanal there are a couple of specialist butchers who care about feed:
I don't do gyms, so I can't help you there. But the American-style gym shop industry is as strong as you'd expect in a country where McDonald's delivers 24/7 nationwide.