Great choices by everyone, I think most of the prolific ones have been named. Maybe All Hallow's Eve, Terrifier, and Terrifier 2 as a prominent trilogy that hasn't been mentioned? (I've personally only seen AHE).
I have to agree with Ozzy about Romero's Dead series, the Evil Dead series, and the Alien series. These three series shaped me the most from a young age, at least.
For the Dead series, I know it's technically no longer a trilogy, but it was for 30 years. While I do think Day of the Dead is worse than the first two movies, it's more about those first two movies being untouchable. It's a good but flawed movie, and the disgusting gore in that film lives rent free in my mind...but so does some of awful dialogue. I am probably also in the minority by saying I like Land of the Dead, which I saw in theaters as a teen. Its practical effects are also disgusting, maybe even on par with Day of the Dead. However, it has horrible CGI and plot issues, though I get what Romero was going for with his commentary. Let's all pretend Diary and Survival don't exist, okay?
The Evil Dead series is probably the one that doesn't have an asterisk around it as far as being a trilogy. The new movies came out 20+ years later and are basically reboots carrying the franchise name; they don't continue Ash's story. What makes this trilogy so great is each movie is completely different, and there seems to be no consensus as to which movie is favored by fans. The only catch is the third movie, Army of Darkness, is an adventure comedy with light horror elements.
Even though Alien: Resurrection came out five years after Alien 3, I still consider the first three Alien movies a trilogy. They work as a clear beginning, middle, and ending to Ripley's encounters with the Xenomorph. Aliens definitely blurs the line between action and horror, but I still consider it to be in the horror subgenre. The film has a lot of tension as the remaining survivors box themselves in from the aliens, repeating a lot of the same failed tactics as the dead LV426 colonists. Alien 3 is a huge let down compared to the first two movies, but it's still watchable. The problem with a lot of these horror trilogies is there will be an abomination like Exorcist II mixed in; Alien 3 is a masterpiece by comparison.