Hi Jetsetkiwi,
I myself have recently been in a situation similar to yours, travelling PER-SIN-LAX-SIN-SGN-SIN-PER. When I was checking in, I had a relative with me to sign the UM form as required. The Unacompained Minor Assitance Waiver Form you're talking about has to be signed by your mother in BNE and handed into the SQ office like you said.
When leaving Phuket, I assume that your father would have to be with you at the airport to fill in the necessary forms as required. Furthermore, if you/your mum purchased your ticket with a credit card, the check-in agent may ask you for the last 4 numbers or the CCV(?) number at the back of the credit card for verification. Thats basically it, and if you can adhere to those 'criteria' listed above, you should be fine.
Sorry I couldn't further elaborate on this situation as I only explained to you my recent experiences .