Getting married in Singapore

Did you visit ROM website? It has defined very clear, what are the requirements before getting married. If you still have doubts or not clear then better call them or visit their office to clarify your doubts. Good luck

I mean do they run background checks on everyone that files a notice?

They can get the data from ICA & MoM (if you are a foreigner and works under work pass).

FYI, all the Govt agencies are in line with the internal arrangements that they can share personal data of any individual as and when needed for verification purposes.

So what if I'm still married in my home country (annulment ongoing) and I want to get married here in Singapore with someone else?

I’m not an expert for this question. You should reach out ROM.

Beware, any false statement or declaration will be landed you in deep trouble here. Avoid at any cost.

As far as I know (but this should be stated on their webpage), a "certificate of no impediment to marry" is required from your home country (and potentially other countries you lived in the past). That certificate would not be issued oif you are already married there.
Bigamy is a criminal offense in almost all countries and should be avoided. Better wait until your current mariage is properly divorced!