How neccessary is a car in Singapore

Out of all cities in this world, Singapore is certainly the place with least need (and highest cost) for a car.
Public transport is excellent and taxis are cheap. You won't even need a rented car (except for trips to Malaysia - if you really want to drive there yourself)!

most condos provide at least 1 complimentary parking space but you should check this with the housing agent or landlord.  If your workplace, home and/or school is near to a MRT station, you probably will not need a car.  Having a car will give you more convenience and the freedom to move around but it is a more costly option then taking public transport.

For most people, not necessary. The MRT system is great but there is also the bus system which only takes a tiny bit longer to work out - and both use the same ez-link ticketing system (stored value card, tap in/tap out).

HOWEVER you may consider one if your kids do a lot of extra-curricular activities, especially if you're not planning to employ a helper.

Shouldn't be too hard to arrange a parking space at home. Parking at your destination is another matter - that can really add up although apparently those with cars know tricks to keep it under control. Hope you're good at reverse parking in tight spaces