Tax-free NRE, taxed NRO and dividends
Singapore generally does not tax an individual on foreign-source income, so your Indian interest and dividends are not taxed in Singapore, and the whole question is Indian, turning on which account the money is in. Interest on your NRE account is exempt from Indian tax under Section 10(4) for as long as you are a non-resident, so it comes to you completely tax-free, and there is nothing to reduce, it is already at zero. That exemption is domestic, not a treaty benefit, so it stands on its own.
The taxed items are NRO interest and dividends. Interest on your NRO account is taxable, and the bank withholds TDS under Section 195, the non-resident section, at around 30 per cent with surcharge and cess. Dividends on Indian shares are taxable too, withheld at around 20 per cent. Those domestic rates are higher than the treaty allows, so the job is to bring them down to the treaty rate.
The treaty rate, and the certificate that unlocks it
The India-Singapore treaty caps these rates below the domestic withholding. Interest is capped at 15 per cent, reduced to 10 per cent where it is paid to a bank or financial institution, and dividends at 15 per cent for an individual shareholder. So your NRO interest can come down from around 30 per cent to 15, and your dividends from around 20 per cent to 15. The treaty rate is all-inclusive, so no surcharge or cess is added on top of it.
To get the lower rate, you give the payer, your bank or the company's registrar, a Singapore tax residency certificate issued by the Singapore tax authority, together with Form 41, formerly Form 10F, and a declaration that you have no permanent establishment in India. One feature of the India-Singapore treaty to be aware of is its limitation-of-benefits clause, an anti-abuse rule aimed at shell and conduit companies routed through Singapore; a genuine Singapore-resident individual with real substance generally meets it without difficulty, but it is why the treaty is stricter than some others about who can claim. Under the new Indian law the relief provision is renumbered, but the mechanism is unchanged. Where too much has already been withheld, we reclaim the excess through your Indian return. That is the whole job, and it is the Indian side we handle.