Answer
No you can't because the Singapore government requires user registration (filling up a registration form and presentation of IDs) of all SIM cards including prepaid SIMs.

I would suggest that you just by a Globe Prepaid SIM or Smart Buddy Sim. You have the option of buying a Globe OFW SIM or Smart Pinoy SIM. Both SIMs are preregistered for roaming and it can remain in roaming status in Singapore as long as you like. These roaming SIMs (mainly targeted to OFWs) allows text from the Philippines to cost only P1.00 per 160 characters.

Please visit a Smart Wireless Center or a Globe Business Center anywhere in the Philippines for more information.

You may also call Globe at 730-1000 or Smart at 888-1111 when calling from Metro Manila.

However, if you prefer to have a Singapore prepaid GSM SIM card you can buy upon arrival at Singapore Changi airport or the Budget Terminal. You buy them at the forex counters. I would recommend you buy Singtel Hi Card (prepaid). Text from Globe to all Singtel prepaid is only PHP1.00 and from Singtel to Globe its only SGD0.05