Sole proprietor would be the lowest cost. You can register your company's name with ACRA in which they will respond to you with the approval or rejection of the company's name.
It's a small fee to pay, you can check their website or go down to ACRA to register.
As far as I know the main difference between sole proprietor and Private Limited is that:
For the former its' liabilities are your personal liabilities, means you have to risk all your asset to cover all the debts when you go bankrupt.
For the latter you will benefit from limited liability for debts incurred. This means that, should your business get into difficulties financially, you and the shareholders of your company would only stand to lose what you originally invested.