I'm a Russian citizen living in New Zealand. I will be flying to Russia, transferring in Singapore. As far as I've found out, I need a visa to enter Singapore? Is it a tourist one? And I'm waiting for my new passport, which may arrive just before I will be leaving. So if I get transit visa in my old passport, can I bring both new and old passports and transfer through, or does it have to be in my new passport?
Singapore allows visa-free airport transit. Moreover, if you want to have a layover in the city, Singapore offers 96-hour visa free transit facility (VFTF) for Russian passport holders. Also see Timatic.
As to visas, Singapore issues them electronically, they are not stamped in your passport. You may apply for a visa as soon as you know your new passport number.