The Shulhan Aruch (O.H. 308: 39), based on the Gemara (Shabbat 128b) rules that animals are not considered vessels and are thus considered Muktze Mahmat Gufo (intrinsically Muktze, and thus not designated for use on Shabbat). It should be noted that like in the case of all Muktze, animals cannot be carried or moved, but they can be touched.
Rabbi Eliezer ben Itzhak (Or Zarua Hakatan, 91) writes that small animals such as Teacup Yorkie dogs whose purpose is to be handled and played with have a purpose and would not be considered Muktze. The Rosh (Teshuvot 82) writes that although he concedes that such animals appear to have a use to people, no distinction is made Halachically and that all animals are considered Muktze. The majority of the great Poskim concur with the Rosh’s position. Rabbi Shmuel Wosner (Shevet Halevi I:62) discusses removing a dead fish from an aquarium on Shabbat and says that since the dead fish is suitable as dog food it would not be considered Muktze and could be removed.
The Shulhan Aruch (ibid:45) goes on to also write that balls are considered Muktze since, according to the Tur (O.H. 308), they serve no purpose, and thus may not be played with on Shabbat. Indeed several Poskim rule like the Shulhan Aruch. Nevertheless, it appears that the Mishna (Kelim 23:1) refers to an improvised type of ball made from rags and does not have the status of a vessel. Nowadays, however, there is a whole industry dedicated to the manufacturing of balls for sports, exercise and the like and these balls serve a very substantial purpose. Because of their importance, balls nowadays would ostensibly have the status of a vessel which is subject to the laws of Tuma and Tahara, and thus would not be considered Muktze Mahmat Gufo. Indeed, Rabbi Yosef Shalom Elyashiv (Ashre Haish, 213) who is also quoted by the Sefer Shalmei Yehuda (pg 91) rules that a ball is not considered Muktze, even for Sepharadim. Rav Ovadia Yosef (Hazon Ovadia, Shabbat 3 pg. 99) concurs with this and writes that one is allowed to give the balls to children to play with as long as they are not used on soil where they might create holes.
Summary: Pets are considered Muktze and may not be carried on Shabbat. Balls are not considered Muktze.