Don't know about Singapore, but they do it the same way as taxis the world over - they either find some place that's out of the way of traffic, like a side street or driveway (especially if the stop will take a while)... or they just sit there disrupting traffic and make people deal with it. Usually it's either a multi-lane road, or a smaller low-speed road where traffic can slow down a little and squeeze by. Not strictly ideal, but people cope.
The taxi stops are there simply for the convenience of drivers and people looking for taxis. Taxis can still stop and pick up passengers anywhere cars are allowed to stop (double yellow line).
I wonder how Uber drivers get away stoping everywhere they feel like. It is somewhat of a traffic disruption.