Overrated: I saw a very similar question recently which prompted me to look up popular Star Trek ships, and I answered Janeway/Chakotay. But while I was looking I also saw that the most popular ship from The Next Generation is Jean-Luc Picard/Beverly Crusher, which disappoints me. It's not surprising because it's a canon relationship that the writers clearly wanted us to care about, but it never goes well - Beverly being Jean-Luc's employer causes constant conflicts of interest, and in Star Trek: Picard it's revealed that Beverly hid the fact they had a son for 20 years. I also think we got enough of their relationship in canon, and would rather see less common pairings that aren't canon or are underexplored.

Favorite: I'm still gonna say Kirk/Spock.