The Shulhan Aruch writes that it is forbidden to ruin or destroy pieces of bread that are the size of a Kezayit or larger. Leftover pieces of bread that are smaller than this size may, strictly speaking, be discarded, but the Shulhan Aruch adds that this is not advisable, because it could lead to poverty. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) explains that this refers to discarding the pieces of bread in places where they will be trampled on, as this is a sign of disrespect. Leftover bread may, however, be discarded in a way that is not disrespectful. And thus Rav Yosef Shalom Elyashiv (1910-2012) ruled that it is permissible to place leftover bread even pieces larger than a Kezayit in a bag, and then place the bag in the trash, as this is not disrespectful to the bread.
Summary: Leftover bread even small pieces must be discarded in a respectful manner. Therefore, rather than being thrown directly into the trash, they should first be placed in a bag which may then be placed in a trashcan.
