I will have a stopover in Singapore in September (ca. 7 a.m. to 7 p.m., not incl. travel to the city centre, so, 12 hours...). - What are the 'must see and do' things there?

Check out their national dish

 

https://foursquare.com/top-places/singapore/best-places-black-pepper

I hope you're able to visit as much as you can for a short period of time. Spending 12hours at Changi Airport alone is not enough because it offers a lot of activities and attractions to do and see such as Butterfly Garden, Canopy Park, Crystal Garden, Bouncing Net, Cactus garden and many more.

Also, I was checking if there are any threads about things to do or must-visit attractions in Singapore but couldn't find any. Thus, here are my suggestions in case someone will be searching for it in the future. Hoping I could be of help.

With the limited time I had during the trip, I was only able to visit these: 
- Changi Airport 
- Orchard Library -
- Gardens By The Bay
- Fort Canning Tunnel
- Marina Bay Sands
- Chijmes
- Haji Lane
- China Town

- Little India

- Merlion Park
- USS

Singapore has so much to offer but if you have limited time, you may consider a private tour https://gowithguide.com/singapore to customize your trip and never miss the city highlights of Singapore. Though, mostly local suggested to spend 2 weeks in the country to enjoy and visit all the attractions.

As its before 2pm and the mongers haven’t yet woken up it may perhaps be too early to mention the ‘Four floors’....