[With emphasis on product liabilities]
Is there a standard list that shows the type of insurance online software companies need?There is no such list, or if someone made it, it would be bogus. There is no one-size-fits-all answer to this. If you want a good answer, then you should go see a lawyer, a insurance specialist, and some entrepreneurs in your area of business, and then make a business decision yourself. Some of the input factors into this decision would be:
Having said that, if it is a lower-risk service (low price, less important user data, standardized and well-shielding terms of service) and the service is provided by a corporation (a legal persona different from your person), then I would personally be OK with operating without any insurance for a while, until the business model is showing good traction. Just don't forget to clean up this, i.e. get good counseling and change things if needed once you begin to have some revenue. This can be tricky, because if you're getting business traction, then you will typically also be very very busy, and may not find the time to revisit legal matters.
Talk to a decent (hard to get, I know) insurance agent. How did you get the idea you do not need liability insurance in the first place - it is pretty obvious that anyone running a business is liable for neglect etc. - whether online or not.
What you may need is:
Turn it as you want, a company is a company.
In the United States, I would encourage you to look into Technology Liability insurance. It is relatively new, but this type of liability insurance addresses the needs of online businesses.