The Gemara Shabbos 128b says that one is not allowed to move animals on Shabbos. In the Halachik jargon, animals are considered to be like sticks and stones (Muktza Machmas Gufo) which are not permissible on Shabbos. Being that there is no conceivable use for them on Shabbos it is therefore Assur to utilize them.
This ruling is quoted by the Shulchan Aruch O.C. 308:39 and most later poskim, and no distinction is drawn between farm animals and household pets. Some pokim (Kaf HaChaim and others) even include “playful” animals in this prohibition.
Other Poskim however DO make a distinction between farm animals and household pets. Their opinion is that a pet is considered like a household item, similar to a toy or a picture, and therefore is not classified as muktzeh.
A proof for this, is mentioned in the Gemara when talking about when a person’s hands are wet on Shabbos. One of the remedies listed is wiping them on an animal’s tail.
If people choose to be lenient in this matter, they definitely have whom to rely upon(R' SZ Auerbach).
(R' Moshe Feinstein had conflicting views which requires a deeper analysis)
The Mishna Brura 305:70 and the Chazon Ish say that if a pet is in distress, one may be lenient and move it or carry it.
* Nothing that is said in this post should be done LaMaisa without first discussing it w/ an official Rabbinic Authority.