First, we should mention that if one plans carefully, there should be little unusable sizable pieces of bread (or other foods) remaining at the end of a meal. Leftover bread can be frozen, used for breadcrumbs (while taking care to avoid meat/milk issues), or left for birds. However, this is often difficult to arrange after semachot, since there can be half-eaten rolls remaining, etc.
There are clear halachot stated in the gemara[1] and poskim[2] regarding the “respect” due to food in general and especially bread. Two related issues are involved: not causing food to be wasted and not degrading food.
One is not allowed to involve food in non-eating activities in a way that is likely to soil it to the point that it will become unappetizing.[3] It is forbidden to throw any food that could get soiled upon falling, and it is forbidden to throw bread even if it will not become soiled, due to bread’s extra importance.[4] The gemara in Berachot[5] explains that the reason for Beit Shammai’s opinion that one should clean the eating area before doing mayim acharonim[6] is concern that the water might fall on and ruin the food. Beit Hillel is not concerned about this, because those in charge of clearing the table will know they should remove k’zayit-sized pieces of bread. We are not troubled about smaller pieces, since, as Rabbi Yochanan says, these may be destroyed. Based on this, sizable pieces are due respect, while small ones are not. This is indeed how the Shulchan Aruch[7] rules.
However, the matter is complicated. The gemara in Shabbat[8] states in the name of Rabbi Yochanan that one may not destroy even pieces of bread smaller than a k’zayit. Furthermore, the gemara in Chulin[9] states that failure to be careful with small pieces of leftover bread may result in poverty. To reconcile the apparently contradictory gemarot, Tosafot[10] maintains that our text in the gemara in Shabbat is incorrect, as the gemara in Berachot clearly says in Rabbi Yochanan’s name that one is not required to care about small pieces. Tosafot[11] further reconciles the gemarot in Berachot and Chulin by arguing that even if there is no prohibition to disgrace small pieces of bread, it could still cause poverty, and this is how the Shulchan Aruch[12] rules. The Magen Avraham[13] distinguishes between different types of lack of care: One is not required to preserve small pieces, but he may not disgrace them, e.g., by having people trample them. Water falling on smaller pieces of bread, thereby making them unusable, is not sufficient disgrace to constitute a problem, whereas larger pieces must not even get soiled by water. The Pri Megadim[14] claims that according to the Rambam, there are no halachic limitations on small pieces, although perhaps there is a danger of poverty.
If one has decided with reasonable cause not to eat some large pieces of bread and there is no issue of wasting them, what should one do with them? Presumably, one should discard them without disgracing them, but what is considered a disgrace? Is putting them in the garbage, the normal place to discard things, a disgrace? Every written source I found on the topic[15] states (without citing classical sources) that one must put k’zayit-sized pieces in a bag before throwing them into the garbage, and many people, especially in Israel, are particular about this.
Is there any explanation for why most members of the American Jewish community where I grew up are not careful about this? We have written in the past[16] that Rav Yisraeli ruled one may put food with the sanctity of Shemitta in a bag and place it in the garbage, even together with foods without sanctity, as long as those other foods are not spoiled. Touching and even getting a little soiled by other foods before being thrown into the local garbage dump may not be considered a disgrace. Based on this ruling, one could claim that since most kitchen garbage bins contain just leftover food, cans, and other items that are not disgusting, putting bread in them may not be a disgrace. However, the easier position to justify here, in regard to both Halacha and avoiding poverty, is to put bread leftovers (especially, larger pieces) that cannot be salvaged in a separate bag before putting it in the garbage.