As an outsider from NYC, I always thought the Sox were the most popular. And they are, but I think it has to do with the season and success of the team.
In the summer, obviously the Sox are the most popular, unless a team is coming off a championship. If the Sox are out of it, like they most likely will be this year, the Pats will be come September.
All I know is, I went to Boston in the end of January, and EVERYTHING was about the Celtics and Bruins. Nobody talking about the Sox or the Pats (or Super Bowl for that matter).
To me, as far as overall popularity, I have the Sox at 1. Then I'd rank the Pats and Celtics in terms of success. Up until the mid '90's, the Celtics were more popular. From then on, up until 2007-08, the Celtics have been.
Put the Bruins in the Finals, and they could more up.
Every city, every a great sports-town like Boston, care more about the good teams then the bad ones. Several years ago, the Celtics fans were in the TD Garden chanting MVP for Kobe. Now, every game is sold out.
I remember going to whatever Yankee game I wanted to in the early '90's. Now, every seat is accounted for, except for a few outrageously overpriced ones.
The Yankees are probably the most popular American team in the world. 20 years ago they weren't nearly as popular.