I'm an Indian citizen and the Singapore consulate accidentally issued my business Visa with my old passport number. Can I still travel without getting a new Visa?
Check with the Consulate
Like Relaxed said, if the consulate made a mistake in issuing the visa then they should remedy their error. In case you made a mistake, then they might be less accommodating. Nevertheless, the thing you should do before travelling is to contact the Singaporean consulate in New Delhi and ask them about your situation.
Transfering the Visa to a New Passport
The website of the Singaporean consulate in Shanghai mentions the possibility of transferring a visa from an old to a new passport. This is done by filling form 14A and sending it to where you applied for a visa. Quoting from the linked website:
Q: How do I transfer my valid visa from my old passport to the new one? A: For transfer of visa, please submit your old and new passport (original and copy), the valid visa, and the fulfilled Form 14A and your 2 inches colored photos to our Authorised Visa Agents.
Whether this is valid solely for Chinese applicant/residents is not specified.
Can I Travel?
I would avoid travelling with the visa on the expired passport. Especially if the visa approval and validity dates are after the passport expiry date. I feel you'll have a hard time explaining the situation to the immigration officer once you land in Singapore. Rather, I'd get this sorted before travelling.
If the passport expired after the visa was issued, then the common practice is that you bring both old and new passport with you when seeking entry into the country. However, the fact that the website of the Singaporean consulate in Shanghai mentions transferring a visa from old to new passport might mean that this will not work for Singapore.