What you are looking for are
1) Fire Insurance: This is very cheap because it cover the basic structure of the building only. It is mandatory to buy if you stay in a HDB flat. If you stay in a
condo/house, I don't think its mandatory to buy. However, if you have a mortgage loan, all (if not most) banks will require you to purchase this.
2) Home contents: Basically, it covers your home furnishing, contents, and some will cover personal liability, i.e. the
property damages you or your neighbour create.
These are known as General Insurance.
AIA, Aviva and Axa are mainly Life Insurance companies. Their premium rates won't be too competitive because general insurance is not their core business.
Assuming you take up a loan from a bank, you can ask them for insurance referrals as well. They usually have a cross referral scheme but depending on the banks, some of their referrals are not necessary good & cheap insurers.
Personally, I would suggest for you to contact a general insurance broker (google and you can find a lot) and get them to source the best quotes for you. Its like buying car insurance, find a general insurance brokers to get you the cheapest car insurance in Singapore.
General insurance brokers (in Sg) can represent a few general insurance companies, hence you don't have to do so much leg work. Of course, the broker get a commission for what they are doing.