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In “The Halochos of Muktzah” page 118 Rabbi Bodner says that animals are muktza. On page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

So you see that animals can be touched as long as that does not make them move.

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it should be permitted, but CYLOR.

Edit: later I found Halachic Perspectives On Pets, an extensive article by Rabbi H Jachter.

He discusses putting a leash on an animal in respect to the prohibition of trapping an animal on Shabbos and says,

A practical application of these halachot occurs when one removes the leash of a pet to allow the animal to run freely in an open area, one would be permitted to reattach the leash on the pet on Shabbat and Yom Tov only if the animal is obedient to its owner. However, if the animal is somewhat disobedient to its owner and resists its owner, it is best not to reattach the leash. It is proper not to remove the leash of such an animal in an open space in order to avoid the necessity of relying on the lenient opinion.

He does not mention muktzah there. Of course trapping is a potential Torah prohibition and muktzah is Rabbinic; so he could be expected to mention the more severe prohibition. But since he does not mention muktzah, he may hold that as long as the person only touches the animal without causing it to move there is no issue of muktzah.

In “The Halochos of Muktzah” page 118 Rabbi Bodner says that animals are muktza. On page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

So you see that animals can be touched as long as that does not make them move.

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it should be permitted, but CYLOR.

In “The Halochos of Muktzah” page 118 Rabbi Bodner says that animals are muktza. On page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

So you see that animals can be touched as long as that does not make them move.

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it should be permitted, but CYLOR.

Edit: later I found Halachic Perspectives On Pets, an extensive article by Rabbi H Jachter.

He discusses putting a leash on an animal in respect to the prohibition of trapping an animal on Shabbos and says,

A practical application of these halachot occurs when one removes the leash of a pet to allow the animal to run freely in an open area, one would be permitted to reattach the leash on the pet on Shabbat and Yom Tov only if the animal is obedient to its owner. However, if the animal is somewhat disobedient to its owner and resists its owner, it is best not to reattach the leash. It is proper not to remove the leash of such an animal in an open space in order to avoid the necessity of relying on the lenient opinion.

He does not mention muktzah there. Of course trapping is a potential Torah prohibition and muktzah is Rabbinic; so he could be expected to mention the more severe prohibition. But since he does not mention muktzah, he may hold that as long as the person only touches the animal without causing it to move there is no issue of muktzah.

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In “The Halochos of Muktzah” page 118 Rabbi Bodner says that animals are muktza. On page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

So you see that animals can be touched as long as that does not make them move.

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it mightshould be permitted.

The same principle would seem to apply to other activities like petting. But, but CYLOR. 

In “The Halochos of Muktzah” page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it might be permitted.

The same principle would seem to apply to other activities like petting. But CYLOR.

In “The Halochos of Muktzah” page 118 Rabbi Bodner says that animals are muktza. On page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

So you see that animals can be touched as long as that does not make them move.

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it should be permitted, but CYLOR. 

1
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In “The Halochos of Muktzah” page 9, Rabbi Bodner says

“Items which are muktza may not be moved or eaten. They may be touched, however, providing this does not cause them to move.”

In “Shemirath Shabbath” Rabbi Neuwirth says 27 (8)

“an animal which needs to be kept on a leash to prevent it from running away may be taken out on a leash ...etc.”

He does not mention putting the leash on. From Rabbi Bodner, I would assume that if the animal would not move while the leash is being attached, it might be permitted.

The same principle would seem to apply to other activities like petting. But CYLOR.