MLB - The Dodgers. It was the 1988 World Series. I was 8 years old and that series made me fall in love with baseball. I also like the Cardinals because my dad likes the Cardinals. I live in the DFW area, so I root for the Rangers when they don't play one of those teams, mostly because I've been to so many games and my son loves them.
NBA - The Grizzlies. It was the Bulls when I was a kid because of Michael Jordan. After Jordan retired, I didn't really care about the NBA until Memphis got a team. I really loved those Grit N Grind (Tony Allen, Mike Conley, Marc Gasol, Z-Bo) Grizzlies teams. They're looking really good this year too. Legitimately one of the best in the league.
NHL - The Predators. Nashville got a team and it's unique. Simple as that. I didn't really care anything about hockey before that.
NFL - Honestly, this one is a little complicated. I had family that were Saints fans and family that were Cowboys fans. The Cowboys started being really good when I got to be interested in the NFL, so I adopted them. Then the drugs, assaults, etc. started happening and it kind of shattered young Dawgg's innocence and I kind of never forgave the Cowboys for that and I became an active Cowboys hater. About this time, the Oilers moved to Tennessee, so I adopted the Oilers (now Titans) and rooted for both them and the Saints (though the Titans a little more because they were closer and had Steve McNair). As more Bulldogs started becoming NFL regulars, I started rooting for them to do well regardless of the team chosen (for example, I would root for the Lions defense to do well because they had Slay). When Dak entered the draft, there were two nightmare scenarios: Browns and Cowboys. The Browns because they've chewed up and spit out more QB's than Bama's D-Line and the Cowboys because, well, I was going to have a tough decision to make. Of course, he got picked up by the Cowboys and rooting against the Cowboys would be rooting against Dak since the QB is the lightning rod for the team's criticisms, and, basically I'll never root against Dak, so I root for the Cowboys, the Saints, and the Titans.
MLS - I guess Nashville SC. I've only watched a handful of MLS games ever, but I would root for them if they were in a position to win something of note.
WNBA - I've taken my kids to some Dallas Wings games. They're cheap and entertaining and they're in UTA's arena, so it's a low maintenance evening. I also root for Vivians and McCowan on the Indiana Fever.