All varying acronyms of incorporation aside, what exceptions (i.e. particular states) are there to the assumption that from state to state, if I need to register my company (for whatever particular purpose), I will be registering with the Secretary of State?
There are many requirements which may apply when you create a business, and they will vary from state to state.
There is a comprehensive list, including foreign countries. (Scroll down a bit to see the state-by-state directions for the United States).
However, every state has different rules about starting a business and what forms of registration are needed. A good place to start is business.gov's Register a Business page. Many states maintain their own publications for how to start a business listing all the registration requirements.
It's never very hard to do (you won't spend more than one day doing research and filling out forms).
That is true for MN and I believe is in general true. Each state though has its own constitution and so could be organized slightly different.