We see from Tos on 45b and Mishna later on 128b and Shulchan Aruch (O"H 308:39 that animals are muckza on Shabbos due to the fact that we have no use for them (can't shecth them or milk them or take eggs from them etc.) Does this pertain to our modern day pets which do have a use on Shabbos? Why?
Tosefos 45b reject the opinion or R' Yosef that a live chick should not be
Muktzah since it can be a plaything for a child. Based on this, Igros
Moshe O.C.4-16 forbids moving a goldfish bowl.
There is a Machlokes between the Or Zarua, who rules that birds which are
kept as pets are not Muktzah, and the Rosh, who holds that they are. Rav
Ovadia Yosef in Yabia Omer 5-26 rules like the Rosh. Shalmei Yehuda Chap 13-11 rules that pets are Muktzah.
In the appendix by Rav Chaim Pinchos Scheinberg, #25, Rav Scheinberg writes
that goldfish would not be Muktzah since they are a species whose only use is decorative, and they are like any other Kli Tashmish of the home that is not Muktzah. I asked Rav Scheinberg if the same logic could be applied to
other animals which are bred - specifically to be pets like dogs and cats. He said that it could be applied however the Minhag ha'Olam is to treat pets as Muktzah and therefore one should not pick them up on Shabbos.
Yours,
Ilan Segal