My advice is to do a degree you love to do. Don't just spend money on a degree for the sake of it, and drag yourself to lecture hall for the next 3 years. If your passion is flying, then do a flying-related degree. If flying is what you really want to do, the degree serves as an alternate, if things doesn't go according to plan, at least you have something to fall back on. So choose one that you know you'll love, even when you're on the ground. If your bank accounts allow, go do a degree with some flying component in it (e.g. Massey, UNSW etc), so that you have some flying experience. Years ago SQ seems to favour those with no flying experience, but I've seen a few, even CPL holders, joining as cadets. They get some hours exempted from flying so they can finish their course a little sooner.
I don't think there's any degree that really help in the application process. We've seen cadets from all sorts of background - bankers, engineers, lawyers, doctors... etc etc etc. It boils down to how you sell yourself in the interview. There were even a few with no degrees. Just a basic diploma, but these cadets have extensive background in aviation e.g. air traffic controller, air force weapons officer... etc.
If you don't know what to do at this stage, while waiting to hit 26, go be an ATC. You learn a lot about aviation, and if you can handle the stress level in the tower, you SHOULD be able to handle the stress inside the cockpit.
Good luck.