שבת קנה:
"נותנין מזונות לפני כלב ואין נותנין לפני חזיר"
The [או"ח שכד:יא] שו"ע paskens based on our Gemara, that since feeding animals is a טרחא, one is only permitted to feed his domesticated animals/pets that rely on him for food. However, any animal that can fend for itself, feeding it would be considered a טרחא יתירה & אסור to do on שבת. Accordingly, רש"ז אויערבך זצ"ל [שולחן שלמה] says one may feed fish on Shabbos at home in a fish tank, since they are dependent on him for their food. The [שם:לא] משנה ברורה brings from the מגן אברהם that one may even feed a dog that doesn’t belong to him on Shabbos, because it’s a "קצת מצוה". This can be understood from our Gemara that says Hashem has pity on the dog because people don’t usually feed it, so Hashem made it take 3 days for their food to digest so they stay full longer. So too, we should follow in Hashem’s ways [והלכת בדרכיו] & feed dogs. The מגן אברהם [שם] says there are those who are נוהג to feed birds on שבת שירה. He says this is wrong since they are not dependent on us for their sustenance and it is אסור to feed them on Shabbos.
The [שכד:ג] ערוך השלחן defends this minhag & says that we are really feeding the birds for our own sake. Since the birds sang שירה על הים we are "מחזיקין להם טובה" . Accordingly, our intention is to remember the זכות of the "שמחת שירה על הים". [תקפג] רמ"א says on ראש השנה we go to a body of water to say the posuk"ותשליך במצולות ים" . No mention of throwing bread into the water. The M”B says that he noticed that when Rosh Hashana falls out on Shabbos, people go on the second day. Maybe, this is because they carry their siddurim with them. Again, no mention of carrying bread. The [תקפג:ה] מחצית השקל says that throwing bread to the fish by תשליך (on שבת) is a מנהג גרוע ואסור. He quotes our halacha above, & says it is obvious that throwing bread & feeding the fish that are in a lake or river would be assur to do.