- Trilogies of declining quality ( 1 > 2 > 3)
This is usually typical of a series where the first movie stands on its own and makes a lot of money, then a few sequels are tacked on to play off of that. I'd think of the Matrix "Trilogy" in this category, Pirates of the Caribbean, probably Back to the Future as well.
- Trilogies with a strong first movie, and a weak middle movie(1 > 3 > 2)
I think of the Mission Impossible movies in this category, the first movie was excellent, the second film was a Matrix/James Bond clone, and the third movie did a good job of getting the series back to its roots.
- Trilogies of increasing quality (3 > 2 > 1)
This is the case when the moviemaking or general quality gets better as you go. Typically in trilogies with tacked on sequels the story tends to wear thin as the special effects get better, so this may be more rare. The prequel star wars trilogy seems to fit this pattern, although that may be a special case since it took them three movies to get to the plot. The man with no name trilogy is another possible contender (the good, the bad, and the ugly series)
- Trilogies with a strong third movie, and a weak second movie. (3 > 1 > 2)
Another example of a weak middle chapter that made up for it later. The Indiana Jones trilogy (so far) fits this one for me, the temple of doom didn't have the same intrigue or significance as the other two, and the last crusade was simply great, you can't go wrong with both Sean Connery and Harrison Ford.
- Trilogies with a strong second movie, and a weak third movie (2 > 1 > 3)
The original star wars trilogy is the classic example of this one, the empire strikes back fared the best critically and a lot of people think the series jumped the shark with the ewok-a-thon in Return of the Jedi, even if it is warm and fuzzy when you're a kid.
- Trilogies with a strong second movie, and a weak first movie (2 > 3 > 1)
This one is even rarer, since it's not very often than the first title in a series is the weakest since that's usually the one to justify all the others. I think the x-men movies are a good candidate though, with adaptations from comic-books you have so much source material to draw from you can really make any number of movies before you get the right story, and the first film didn't do as good of a job with the characters as the other two.
I'm still somewhat undecided on how I'd rank the Lord of the Rings films or the Spiderman movies. My intuition is that for most trilogies the first movie is the best, although it may be different if the entire trilogy is conceived in advance, as in the case of Lord of the Rings, or if the trilogy is composed with a two part sequel, as in the case of Back to the Future, The Matrix, and Pirates of the Caribbean.
Update: Rottentomatoes (a site which aggregates ratings from actual critics) rates Spiderman, X-men, and the original Star Wars trilogy 2>1>3, LOTR and the Star Wars prequels 3>2>1, Back to the Future, Austin Powers and Indiana Jones 1>3>2, The Matrix and Pirates of the Caribbean and Spiderman 1>2>3, and Mission Impossible 3>1>2. Wasn't able to find any trilogies that were ranked 2>3>1.
Further Update: The Five Harry Potter movies so far are ranked 3>4>2>1>5, which is probably the order I would rank the first five books as well.