If the surface is not a pair of pants, there are infinitely many different isotopy classes of pants decompositions (which, as mentioned in the comments, can be made compatible with a constant curvature metric).
They are all "connected" by a finite sequence of "moves". The collection of the isotopy classes (thought of as 0-cells), connected by edges (1-cells) if they are related by the aforementioned moves, and with 2-cells added according to relations among the moves forms a 2-complex that Hatcher has shown to be simply connected. See here for precise definitions and the proof.
On the other hand, the mapping class group acts on the 1-skeleton of this complex (see here), and the quotient is a finite graph. So there are finitely many homeomorphism classes of pants decompositions.
As far as I can tell, the precise number of homeomorphism classes of pants decompositions is not presently known. See here for a lower bound however.