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.

up to homeomorphism. It has infinitely many pants decompositionsup to isotopy. It is important to understand the difference between these two concepts. Check out the Wikipedia pages! The hyperbolic metric doesn't effect the above two statements. $\endgroup$topologicalpants decomposition (defined up to isotopy), you can replace the cutting curves by geodesic representatives. In this sense, you can make any decomposition you like compatible with the metric. $\endgroup$