Hi mate, don't listen to the nay-sayers
If you are a serious bike-commuter, then you will love it here.
All the roads are in great shape, there are few hills, and once you get used to the driving patterns, it is quite safe. I bike commuted in Vancouver for many years and find that there are fewer 'heart in mouth' experiences here. You'll always find mental-midgets and visually challenged drivers, but Singapore is certainly no worse, and I think it is far better than Canada. But I have also mellowed with age so I might be a bit less volatile in how I deal with traffic than I was back home 20 yrs ago.
I've been commuting consistently maybe 3x per wk from Clementi to the city (a good, hard 20 minutes) for 16 yrs now and with only a few problems have quite enjoyed it.
The only downfall is the climate so if you ride further than about 50m you will need a shower on the other end.
Really, don't waste your time with a portable bike, get a real one - they slide into the back of taxi's just fine (you have to call for the larger new models with deep boots, but the $3.50 extra is no big deal if you want to move your bike around).
The bike doesn't need to be expensive, remember that fitness is the goal so a moderate $200-500 bike is a much better workout than a $5k model. You can find great 2nd hand deals from expats departing. There is low crime so your bike is not a huge target, but if you buy an expensive model, there is always that fear. I'm onto my 3rd bike, I finally got a nice one (but not too nice) and never had a problem with theft or vandalism.
Get a gym membership near work and use it for the shower, should cost a few dollars a day which is offset by savings on your commute. Buy a small wardrobe or large hanging suit bag for your office, or just take 3-4 shirts in once a wk in the taxi.
If you want there are top shelf bike shops springing up everywhere but I use a small local bike shop that doesn't charge an arm and leg for every visit. My guy always has a cold beer in his fridge and for adjustments and small stuff simply won't take any fee - good business as I'm a loyal customer for more than a decade now.
And mostly keep your head up, eyes on the road, and a smile on your face, it makes dealing with traffic so much easier.
There are a lot of biking clubs and such popping up for weekend rides but I'm just a commuter so don't get involved in the weekend processions, you can find them through any of the higher end bike shops or online forums. Buzz if you have any questions, more than happy to help when you hit the ground.
Cheers,