[sky13]

Singaporean here. It will be hot and humid in Singapore, and recently there have been more than a few thunderstorms (mostly passing by quickly, but thunderstorms can linger for hours sometimes).

Shorts are definitely appropriate for Singapore!
As are other things like flip flops. MBS is a bit more upmarket but doubt they'll bat an eyelid at shorts. (I've gone around the adjacent MBS mall in running gear and not felt out of place, although this was early in the morning for breakfast after a run...) Bring an umbrella (or poncho - but it gets hot inside a poncho) if you intend to be outdoors a lot, in case of rain.

In terms of HKDL and SDL - HKDL is connected to the metro (MTR) system and is on the same island as the HK airport, so you could take the MTR to get to HKDL. Or a taxi. SDL is also connected to the metro system but when I went we just took a taxi from the airport as that seemed to be the easiest and fastest route.

Have fun on your trip!