The caveat, is you must operate like a business - including proper insurance levels. If you're not acting like a business, then you don't get business-like protections.
Source: Single member LLC who has discussed the issue with my lawyer.
In my experience, insurance levels are determined by the work you do and clients you have, not by your entity type. Most of my 1099 contracts these days come with a professional insurance clause that is independent of the kind of business I run.
This is like asking "what insurance do you have?" Some people will carry health, auto, life, home, flood, and earthquake insurance. Some people opt for some or none, depending on applicability.
An insurance broker will happily talk you through what each gives you, and what is most relevant to your co, but it will almost always include GL (general liability), E&O (errors and omissions), and some sort of "cyber" (to pay for resulting damage from hacking). Expect other flavors of insurance to also be applicable.
The caveat, is you must operate like a business - including proper insurance levels. If you're not acting like a business, then you don't get business-like protections.
Source: Single member LLC who has discussed the issue with my lawyer.