Question: We recently got a pet rabbit. Are we allowed to play with her on Shabbos?
Answer: The Shulchan Aruch (OC 308:39) writes that animals and birds are muktza and so must not be moved on Shabbos (See Shabbos 128b). Tosafos (Shabbos 45b) and the Mordechai (Shabbos 316) write that there is a view that as children can play with pets, they serve a practical use. Nonetheless, they write that animals are muktze, comparing them to figs and grapes which are in the process of drying. Likewise, the Maggid Mishna (Shabbos 25:25) explains that animals are considered to have no practical use on Shabbos (See Mishna Berura 308:146). The Shulchan Aruch Harav (OC 308:78) adds that this prohibition applies even to giving a pet bird to quieten a crying child (See Shemiras Shabbos Kehilchasa 27:n101).
R’ Ovadia Yosef (Yabia Omer OC 5:26) quotes other poskim who don’t consider animals to be muktze but writes that we follow the great rishonim such as the Rosh who disagrees. The Mishna Berura (324:28) cautions that when one feeds one’s animal on Shabbos, one must ensure that they don’t move it at the same time.Nonetheless, the Mishna Berura (305:70) and R’ Ovadia Yosef write that if one’s animal is in distress one can move it if necessary. Thus, while R’ Moshe Feinstein (Igros Moshe OC 4:16) writes that fish and aquariums are muktze, R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 27:28) allows one to return a fish that had jumped out of its aquarium.
In conclusion, one cannot play with or handle animals under normal circumstances on Shabbos.