Yes, there are Olympiads for Maths, Physics, Chemistry and Biology.
I was personally a national representative for Physics Olympiad, so I can comment on this. The information is for JC only.
Basically, schools will choose JC1 students to represent the school for the Singapore Physics Olympiad. The selection can be based on recommendations, your class physics results, maths results, a selection test, and/or a combination of the above.
After you finish the Singapore Physics Olympiad, which is conducted by NUS, NUS will select students based on scores. The top few will be selected to go for a short stint of training at NUS. From there, there will be another selection test to select students to represent Singapore.
There is Asian Physics Olympiad (8 representatives) and International Physics Olympiad (5 representatives).
As far as I know, the same goes for Maths, Chemistry and Biology. I'm unsure if there is a Asian competition for these.
If we move down to O levels, there is also the Singapore Junior Physics Olympiad. This is to start preparing students for the Singapore Physics Olympiad and hopefully and eventually represent Singapore for the Asian and International competitions.
I'm a trainer for junior physics olympiads for 2 schools :)