I am in Virginia. My current trailer was bought new, but my previous one was bought new. As far as i remember, with the used trailer, i just took the bill of sale and the previous registration to the DVM (I do not remember if there was a title). I am sure it can now be done on line.
Maybe things have changed, but no insurance was required, as any liability is covered by the insurance on the tow vehicle. I do have insurance on my current trailer, but it is just “fire and theft”, and that runs about $20 per year.
The only registration I was offered for a trailer was “permanent”.
I am in a county with personal property tax on vehicles. Every year the tax bill comes with the trailer marked “not subject to tax”, but I understand other people do get taxed on their horse trailers. I am not sure of the reason for the difference.
See the Virginia DMV web site
https://www.dmv.virginia.gov/vehicles/#trailerreg.asp
which also has a link to the registration form (which is primarily aimed at registering a car/truck, but does have check boxes for a trailer)