I leave mine unplugged. I don't want to have the Inverter on 24/7 year after year. I do have the battery hooked up to $20 trickle charger outside next to the battery 24/7.
I figure that the Inverter, like any other electrical piece of equipment will eventually break down sooner than later if left running all the time, and it being more difficult to change out and a lot more expensive than a $20 trickle charger to replace someday in the future.
Than their is always the thought that the Inverter could overheat and cause a fire inside the trailer.