Great news! I was married in Singapore as well. You don't have to go the ROM route if you don't want to be married in a church.
Go down to ROM and register. Then, get a copy of the Singapore Government Directory (they didn't use to volunteer this info at the ROM) which you can buy at MPH. Look up the section for Justice of the Peace. Then get one of them to agree to marry you on your selected date.
Meanwhile, sort out where you'd like to have it. We had ours at Raffles Hotel and had a lunch reception. It was gorgeous. Make an appointment, go in, and they will review budget etc. etc. they did everything including flowers for us.
When you've got your location and JP lined up, go back to the ROM and ask for a change of venue (i.e. from the ROM to wherever)
And voila, you're done.
You will need to collect the marriage certificate a few days before the ceremony, and need to bring two adults I/C or passports with you. They didn't tell me this at the time but it turns out those are your two official witnesses and their names are typed on the document. They don't need to go with you to collect, but you need their documents and they have to be present at the marriage.
I don't know if the procedure is different now, but that's what we did.
Good luck and have fun planning!