Thanks again Rollin for all the advice. Here's my report from today:

After asking around here this morning, it looked like East Coast Park was my best bet overall.
I decided to walk from the Hilton to the park, it took me a bit under 3 hours to get there and to the bike rental kiosks. They were all mountain bikes, all of them had seen better days.

The first bike I was given was a bit of a bust, every attempt to shift gears ended up with the chain tangled around the crank.
It was kind of ironic because I only wanted one gear anyway (the top one) for the flat terrain. I took it back and exchanged it, the second bike turned out to be much better.

It cost me 7 SGD (a bit under $5) to rent it for 2 hours. I tried going out on the street, but after nearly getting hit by a police car and me being irresponsible and riding bare-headed (didn't feel like carrying the helmet on my walk) I decided to go back to the park. I'm pretty sure I wasn't supposed to have the bike outside the park anyway.

I rode up and down the park a couple of times, one time through I passed a charity ride (Bike'n'Blade). It was nice having people in orange shirts pointing out the route, as opposed to the arrows on the ground that I am used to seeing. The park seemed like a nice place to hang out, I was impressed by the cable water-skiing and the dedicated mini-race track for radio-controlled cars, I've never seen either of these before elsewhere.

After my ride I was a bit hot so decided to walk over to the MTR and take it back to the hotel, much faster and cooler than my morning walk.

Here's a photo of me and the bike. As I was setting up for this shot, I was bitten on the foot by a spider. I pulled it off and just hoped it wasn't poisonous! (So far so good.)
More pictures here.

Paula