The Boston Red Sox have sold out the most consecutive games in MLB history while the Portland Trailblazers have sold out the most consecutive home games in the NBA (and in North America for any sport).
The Boston Celtics
The Boston Red Sox hold the record with 615 consecutive sellouts as of August 14, 2010.
Barcelona is the team in the La Liga with the most number of consecutive wins. It has won 16 consecutive games.
The Boston Bruins
Boston Celtics 17
New England Patriots
The New England Patriots