Could you please tell me what the Halacha is with respect to saving the life of a pet on Shabbat? If a pet becomes suddenly very ill on Shabbat, may one call a veterinarian? May one take a pet to the veterinarian? If not, what options does one have other than to let the pet die or remain in tremendous pain?

One is not allowed to transgress the Shabbat to save an animal. This includes Rabbinic and Torah prohibitions.(Sulchan Aruch 332, in the Mishna Berura, Shmirat Shabbat Kehilchata 27,55) However, one is allowed to handle the animal, i.e. to pick it up or to relive it of pain ect. One is also allowed aid a non-Jew to deal with the animal. It is also allowed to perform things which are prohibited specifically because they are medically related (SSK 27, 54-6). However, it is advisable to ask a Rabbinic authority if a specific case comes up, because these Halachot are complicated and some Poskim seemed to allow certain Rabbinic prohibitions under certain circumstances (see SSK 27, note 166, 168). In these laws there is no difference between saving the life of an animal or stopping a leaving pain (See SSK 27, note 98). Rabbi Baruch Paz