I'm getting there, slowly but surely:

28th April : Applied for a refund using the online option, as I could see my outward flight departing on the 30th June was no longer on the June timetable.
28th April : Received the automatic acknowledgement.
1st June : Received the official cancellation email for my outward flight on the 30th June.
15th June : Received the official cancellation email for my return flight on the 22nd July.
15th July : Received an email stating the credit card I used to pay for tickets was expired and requested new details, which I supplied on the same day.
16th July : Received the Refund Notice confirmation, stating that the refund should be credited to my account within 4-6 weeks.

So that's taken 11 weeks so far, plus if the refund does take an extra 4-6 weeks to be credited, then that will be around the 15-17 week mark it will have taken, which is close to the 16 weeks that Singapore Airlines originally stated on the website.