If a utility trailer was left abandoned on my property for a year, can I claim ownership now and how?

No, you cannot. I suppose what you could do is notify the owner that they have 30 days to remove the trailer and if they don't, they will be charged $10.00 per day for storage. At some point, the charges may exceed the value of the trailer and you can try and sell based on the charges. Or you could ask the person to sign the title over to you. You will probably need to talk with a local attorney to make sure you do it in compliance with your state's law.