Tn does not require registration on trailers other than RV style trailers. Travel trailers, etc must be registered. I also register my 6x10 aluminum trailer because it heads out of state on occasion and I do not want to be arguing TN trailer registration law with a LEO on the side of the road. My advice , which is worth the paper it is not written on, is even if your state does not require it to be registered, register it and save yourself potential hassles when you do take it out of state. It is not unusual to find LEOs who do not know your state's motor vehicle laws, nor should they have to.
Randy