Patriots.
The Dallas Cowboys hold the record for most postseason appearences with 28. They also hold the record for most postseason games played with 54.
The A team.
Derek Jeter (aka Mr. November) holds the record for most postseason appearances. He's played in 158 postseason games with a .309 career postseason batting average. The New York Yankees have made it to the postseason every year since Jeter joined the team, with the exceptions of 2008 and 2013.
New England Patriots
1977
New England Patriots
usc
Detroit Lions
7
Angels
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).
Internazionale is the team in the Serie A that has the most number of consecutive wins.