How do you search public records/copyrights/wherever else to make sure a name is not already in use?

@Rebecca In the UK we use this:

find-and-update.company-information.service.gov.uk

While that doesn't help you in the states, I'm sure the government over there will provide an equivalent.

https://www.sba.gov/business-guide/launch-your-business/choose-your-business-name maybe?