Non-impediment letter from German Embassy in Singapore

It is correct that German embassies do not issue such a document.
The German spouse should get it from the family registry ("Standesamt") of his home town or the town he/she last resided in in Germany.

Thanks for your reply!

My partner has already got a letter (a formated document) from the town hall and the official from the family registry of the town hall reached out to the regional office in Stuttgart and  they advised that they can't produce such a letter and the embassy should be able to produce such a letter.

The issue is that ROM in Singapore, requires a document that states details of the groom and bride and legalized by the Embassy. We are stuck now as none of the German institution (in Germany or in Singapore) could help us and produce such a letter/document.

We are wondering if there are German nationals who registered their marriage in Singapore faced the same thing, and perhaps have found a solution to this.

I am German and married in Singapore, but that was 15 years ago (and me PR). ROM did not ask for any documents from Germany and in fact not for anything that shows I wasn't married yet. I guess that has changed since - and rightly so!
The embassy has no access to family records, so they certainly cannot issue non-impediment document. (They can, however, legalize documents you get elsewhere!) The family registries in Germany are stereotypical buerocratic nightmares and will certainly not issue anything other than the standard non-impediment letter, which you already have.
I assume the difference between that standard letter and what ROM requires is small, but maybe you can explain this further?
It might help to get a written statement from both the family registry and the embassy that such a document cannot be issued. Submit that to ROM and maybe they will accept the document you already have.
If not, marry elsewhere (Denmark and Las Vegas are supposed to be easy).