You may need to get PSA CENOMAR even if you go through the consulate in Singapore for your CR1 visa. There's not many who report from SG about it, but there's reports here on VJ about the consulates in Hong Kong and Japan asking for CENOMAR from Filipino applicants. But it's not much of a hassle to order your CENOMAR online.
Note also that it's acceptable to present a CENOMAR and not a CEMAR (Advisory on Marriage) even if you are already married. The consulate just wants to check that you were never married in the Philippines. That's what I did -- married in the US, didn't file ROM, presented a CENOMAR (not CEMAR) to the US consulate, got my spouse visa just fine. No questions asked why I didn't file ROM.
That said, my non-filing of ROM was out of ignorance/negligence, not intentional. I realized too late that ROM was required for my PH passport renewal with married name. Now I'm stuck with a passport, green card, and other US IDs/accounts in my maiden name, and it would be a hassle to get my name changed on the documents before I get US citizenship.
So, if you're fine with having your passport list your maiden name, then no need to file ROM with the PH consulate. But if you do want to use your married name in your PH passport, then you must file for ROM before renewing your passport. No need for a new passport for the I-130 petition. Just need one in time for the visa interview. Your spouse visa, green card, and SSN will list whatever name is in your PH passport.