Can you apply for a Schengen visa in Singapore without being a local resident?

NB: Answer nearly fully copy-pasted from this answer by @Relaxed

Usually no, you won't be able to apply for a Schengen visa in Singapore.

The general rule, defined in article 6 of the Schengen visa code is that

1. An application shall be examined and decided on by the consulate of the competent Member State in whose jurisdiction the applicant legally resides.

And you're obviously not a resident in Singapore so you should not apply there. The same article does however open another possibility, namely that

2. A consulate of the competent Member State shall examine and decide on an application lodged by a third-country national legally present but not residing in its jurisdiction, if the applicant has provided justification for lodging the application at that consulate.

If you just want to apply because it's more convenient for you, I think you can just forget about it. Beside exceptional circumstances the only thing I can think of is to argue that it was too early to apply in India (you are not supposed to apply more than 3 months in advance for a Schengen visa and you intend to stay 5 months in Singapore).

But even if you do have a convincing justification, most consulates in Singapore have stricter requirements. For example, here is what's on the website of the Swiss Consulate General in Singapore:

Proof of legal residence in Singapore, valid for at least 3 months beyond the intended date of departure from the Schengen territory. Original plus 1 photocopy back and front of Employment Pass / Permanent Resident card and re-entry permit / Dependent Pass / Student Pass.

So even though they could (or perhaps even should) entertain the possibility that you have a good reason to apply in Singapore without being a resident, they are obviously not very friendly to the idea. (I haven't checked all 20+ consulates in Singapore but I came across similar language from at least half-a-dozen of them in the past.)

Furthermore, nearly all Schengen consulates in Singapore use some third-party outsourcing company to filter visa applications. They will just go through their checklist, notice that you don't have the right Singapore visa and reject your application (this means they will decline to process it, it does not count as a refusal and there will be no traces of this). Merely getting to talk to an actual consular officer who knows the rules and could accept to examine it will probably be an uphill battle.

If at all possible, I would therefore apply in India (or in your current country of residence, if it's not India).

If you positively can't do that and still want to try your luck in Singapore, I would try to take a print out of the relevant parts of the regulation and use it to insist (calmly but firmly) that your application should be examined nonetheless. Also join a short letter for the consulate itself in which you can show that you know the rules (mention article 6(2)) and lay out your argumentation.