  We study the sets of semistar and star operation on a semilocal Pr\"ufer
domain, with an emphasis on which properties of the domain are enough to
determine them. In particular, we show that these sets depend chiefly on the
properties of the spectrum and of some localizations of the domain; we also
show that, if the domain is $h$-local, the number of semistar operations grows
as a polynomial in the number of semistar operations of its localizations.
