I was reading this question, which inspired me to ask the above question. In Star Trek, there have been mentions of baseball, tennis, basketball, water polo, ice hockey, racketball, soccer (in Enterprise in 2152), volleyball, fencing, and other sports, including martial arts. However, we have rarely heard of any kind of professional sports leagues.
In the Star Trek: Enterprise episode "Minefield" (set in 2151), Captain Archer tells Lieutenant Reed that England has made it to the World Cup Final. He also watches a water polo match in the episode "Catwalk" while trying to fall asleep. However, other than this, I don't think I've heard any other mentions of professional leagues. Sulu, Picard, and Guinan show fencing skills, but there is no mention of any professional competition; baseball--which has apparently died and is just played by some enthusiasts and amateurs--doesn't have a World Series anymore; Captain Janeway once practiced her despicable tennis serve for the audience (I play tennis--I know what bad form looks like), but there was no accompanying mention of a top player or a grand slam; and Bashir beat O'Brien handily at darts due to his status as a GMO.
I understand that some professional sports don't exactly match well with the values of Star Trek. This is especially true for brutal sports like American football, in which a player can be severely injured--like the incident that happened just yesterday, in which a football player suffered cardiac arrest on the field. And yet parrises squares, a violent and potentially fatal game, is enjoyed by many youngsters in the 24th century. But there are plenty of other healthy, safe, and competitive sports that people could enjoy.
I know that people in Starfleet don't necessarily have time to keep track of sports when they are busy with daily duties, trying not to start a war with the Klingons/Romulans/Cardassians, or avoiding assimilation by the Borg, but there would be someone who would have the spare time and interest to follow their favorite players or matches.
So do professional sports leagues still exist, or have they all become a much smaller activity than they are now?