I cycled every day for a month along a mix of park connectors (cycle paths) and two and three lane roads.
I only had one close pass from a car because on nearly all roads they have a whole extra lane to use to pass you in so if you ride assertively (in the middle of a lane) they just go around you.
And the park connectors are beautiful, fast tarmac with no cars at all. With some creativity you can link them up to create a great commute to most places. My route home went past a Riverside pub - I was on segregated cycle lanes the rest of the way home so I could easily stop for a beer with little risk.
You do usually arrive soaking wet from sweat or the rain. But that's a minor inconvenience compared to being cold! Biking in Singapore is warm, fast, and safe - much better than most places I've commuted.