We know of two types of punishment: the discipline wand and release.
We learn about the discipline wand in Ch. 7 when the Council of Elders is giving Asher his assignment. Here were learn that as a very small child, three years old, he mixed up the words "smack" and "snack." As a result he was "smacked" with the discipline wand. Although the first few smacks were fairly light and just on the hand, Asher continued to make the same mistake and this led the smacking to increase.
"for each lapse the discipline wand came again, escalating to a series of painful lashes that left marks on Asher's legs. Eventually, for a period of time, Asher stopped talking altogether, when he was three" (Ch.7).
Although everyone later laughed off Asher's mistake, even him, it is clear that this punishment had a very serious effect on Asher as a child.
Release is another form of punishment, and we learn about that much earlier in the book. In Ch. 1 a plane from Elsewhere flies into the community and the whole community goes into a brief panic. It turns out it was simply a pilot who got lost, took a wrong turn, and ended up flying into their air space. However, this is no small error according to this community and over the loud speak they promptly announce, "Needless to say, he will be released." Later in the book Jonas reflects on saying to Asher once that he would be released. He only meant it as a childish threat while they were playing, but he got in trouble for saying it. Release is the most serious punishment the community can hand down and it is no matter to be taken lightly, even by children.
If I remember correctly, the "discipline wand" was used at times, and release, or death was also a form of punishment. Society members considered it to be the most severe form of punishment.