I have a utility trailer not used for a business. Do I list it as property to be taxed on federal or state taxes?

if not using it for business you don't have to report it.

If you live in a state where the vehicle registration includes a personal property tax, the tax on the trailer is deductible, on the federal return. That is, if you are already deducting a portion of your car (or truck) registration, you can also deduct a portion of the trailer registration fee.