Feeding animals:[1]

It is permitted for one to feed home and farm animals, food and drink, if they are dependent on him for their food. It is however forbidden to feed home and farm creatures if they are not dependent on him for their food, such as bees, or pigeons.[2] It is forbidden to even simply place food in front of them.[3] It goes without saying that it is forbidden to feed wild animals or birds.

Dogs: It is permitted to feed dogs, even if they are wild and not owned[4] and are hence not dependent on oneself for their food. Furthermore, it is even a slight Mitzvah to do so.[5]

Pigs: It is forbidden to feed a pig on Shabbos [even if one owns it and it is dependent on oneself for food].[6]

Shabbos Shira:[7] Some are accustomed to place food in front of animals on Shabbos Shira, [even if the animals are not dependent on people for their food]. However, based on above doing so is incorrect being that these animals are not dependent on people for their food.[8] [Practically one is not do so, or even allow children above Chinuch to do so.[9] Rather one who wants to honor the custom is to set up food for the animals from before Shabbos. Likewise, one is to explain to children the importance of being merciful to animals.[10]]

Feeding insects:[11] It is permitted to feed worms [and other creatures] that one owns if they are dependent on oneself for their food.

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[1] 324:7

[2] As they eat food from the field and are not dependent on the owner for food. [ibid]

[3] As the Sages only permitted troubling oneself on Shabbos for animals if the animals are dependent on oneself for their food. [ibid]

[4] Other Opinions: The Machatzis Hashekel understands the M”A to rule that if the dog is not owned it may not be fed.

[5] As G-d had mercy on the dog which finds little food, and made its food last in its abdomen for three days. [ibid]

[6] As it is forbidden to raise pigs, as the Sages said “Cursed is the man which raises pigs”. [ibid]

[7] 324:8

[8] So rules M”A.

Nevertheless some justify the custom on the claim that since their intent is for a Mitzvah, in order to repay the birds which sang Shirah by Kerias Yam Suf, therefore they may do so. [Tosefes Shabbos 117; Minchas Shabbos; brought in Ketzos Hashulchan 131 footnote 5; Aruch Hashulchan 324:3; Nemukei Orach Chaim 324 states it is an accepted custom from the Holy Tzadik Reb M”M of Rimnav, to give the birds crumbs on Shabbos Shira in reward for them having eaten the leftover Man which Dasan and Aviram cunningly spread out on Shabbos night to make it appear as if the Man fell on Shabbos. He then goes on to defend the custom based on Halacha.]

[9] Shaar Halacha Uminhag 1:149

Other Opinions: Ashel Avraham Butchatch 167:6 allows children to do so being that it is not a complete Shvus.

[10] Shaar Halacha Uminhag 1:149

[11] 324:9