Hi,
In terms of boats from Singapore, you're pretty limited to either the White or Black Manta or a cruise with a stop in Redang. The Black Manta is in Indonesia this time of year, but the White Manta has a Malaysia trip 19th - 21st that would work, but I find the price quite expensive, cheapest base price is SG$650 for what you get. Not dove with them though.
Whitemanta.com
Otherwise, you can join a trip to Tioman / Aur / Dayang in Malaysia. Most local shops operate to one of the 3 each weekend. Bus leaves Friday night and drops you back off Sunday night. Usually around SG$400 with equipment and you get in 6 dives max with most. You can also self-organize a trip to those locations, but that's gonna require some more effort and booking your own bus / ferry ticket up. You can see some sample itineraries at
Amazing Dive Travel. Diving is not fantastic at these spots though.
Better Malaysia diving would be to head up via a 10 hours bus from Singapore or 12 hour train ride from Johor to Perhentian or Redang. This is the prime season for this part of Malaysia and the diving is warm and cheap. Best bet is to catch a bus from the company Transnasinoal, straight from Singapore to Pulau Terengganu / Jertah
Transnasional. Book these tickets early, because there's only one Trans bus per night and it fills quickly. The other companies are much more expensive and labor intensive through customs. You can try Turtle Bay Divers / Watercolours and Tuna Bay in Perhentian, all are pretty well regarded.
Best place in reach this time of year is probably Bali. Flying up Wed night gives you 3 full days diving with a day to enjoy the island and catch a late night Sun flight back. You can fly with any of the budget carriers (Tiger/Jetstar/AirAsia/Garuda/LionAir although only a few have evening flights. You can go do the Tulumben area wreck dive and then go Mola Mola hunting down at Nusa, prime time for them, but its deep (30+m) and colder (22Cish) diving. There's also a decent manta dive. Check out Aquamarine's page for an example of the 4 day safari packages.
AquaMarine Diving Bali. Not sure you have time to really get to any of the other prime Indonesian diving (Sulawesi / Raja etc.)
The other often visited spot that is in season is Ko Tao in Thailand. The areas been a bit over dove, but viz should be good and the diving is cheap with the chance of whale sharks. The only direct flight is with Bangkok Airways and its pricey, but you can catch a connection with most of the other airlines. This will be a bit riskier with your schedule.
Other options, although the season is not prime include, Cebu area or Malapascua, which you could just squeeze in or Phuket (take your chances with the rain). Check out Thresher Shark Divers or Evolution in Malapascua.
Others may have some lesser known destinations! Good luck and let me know if I can help any more.
Ryan