To become a White Mountains Trails Finisher you must hike all of the trails in the AMC White Mountain Guide.
Unfortunately, this goal is not as static as completing a list of peaks. The trails are constantly changing, trails open and close, and what does "all of the trails" really mean?
The best place to start is purchasing an up-to-date copy of the AMC White Mountain Guide. This guide gives excellent descriptions of the trails and also points out many tiny side trails to scenic viewpoints. Next you can download one of the spreadsheets provided below. You may use any version but are encouraged to use the most recent.
Note: Steve Smith was consulted for the 29th Edition. That spreadsheet already includes additional spur trails that Steve felt should have been added.
You are expected to make an honest effort of hiking all the trails in the guide. This includes visiting campsites, shelters, and scenic viewpoints that may not be listed in the spreadsheet.
Trails can be traced in the winter. When the snow is deep it is easy to get off trail and you may at times not know if you are exactly on the trail. As long as you've made a good faith effort to stay on the trail you can give yourself credit for that section of trail.
All segments or pieces of a trail must be hiked, walked, run, skied, snowshoed, or butt slide, at least once. And, be done without the aid of any sort of transportation with wheels.
The mode of transportation used to get to each particular segment or section of trail is not an issue. Any combination of other modes of transportation, bike, motor, vehicle, horse, etc., can be used.
In the interest of not becoming too obsessive you are not required to hike herd paths or unofficial trails that you may come across. Visiting a campsite or shelter is adequate enough. You do not need to walk the path to each individual tentsite, platform, privy etc...
Bottom line, this is based on the honor system. If to the best of your knowledge you've hiked all the trails in the guide then you are eligible to be a White Mountain Trails Finisher.