Kirk knew what a movie was when Edith Keeler suggested they go see one (he just didn't know who Clark Gable was) so that form of entertainment must exist in some form in the future. Plus, so much of our movies and television shows now have been preserved by various academies and whatnot that I'm sure they're still around in the 24th century to be viewed. The only reason why we never see Picard watching Citizen Kane, or Sanford and Son is because of the licensing costs. Musical pieces and plays that are in the public domain don't cost Paramount anything to put into the production. As far as them not putting 'modern entertainment' shows into Star Trek would probably just come across as awkward: "Hey Wesley, did you check out the new episode of 'Cardassian Friends' last night? Gul Bing is hilarious!"
Oh the possibilities ...
"The Odd Couple" with one Cardassian and One Bajoran, "Can two divorced humanoids live together without driving each other crazy?"
"The Dukes of Qo'NoS," the adventures of two Klingon cousins in their suped up Bird-of-Prey named the General Chang who every week must outwit the corrupt House of Duras.