How do you find a trainer name that isn't taken in Pokemon Go?

every short or obvious name is gone, so stop trying to get the clean version. names can only be letters and numbers, no spaces or symbols, and there's a length limit around 15 characters, so the trick is combining two unrelated words you actually like rather than adding numbers to a pokemon name. "MossLantern" style stuff is usually free. and pick carefully - you only get one free name change after that.