In my case, which includes raising two children with the oldest now living on her own as a college student and will be getting married soon as well as another daughter who is an 8th grader and 13 years old, it all depends upon the infraction.
If it is a matter of disrespecting another student or a teacher, then phone privileges are taken away for a few days to a week and the child must write a letter apologizing for the unacceptable behavior to both the student and the teacher.
If it comes to homework or school work, I have the teacher send extra work home for the child to complete and again, a letter of apology along with the reason as to why this is not acceptable behavior is included.
I have never had any problems with fighting or anything of that kind but the girls have always known that is something I WILL NOT tolerate in school. If this because a problem, then I take away everything for about a week and we have long talks as to why it is not acceptable behavior, how it can ruin their reputation in school and making it difficult to participate in any events at school. Then, yes waiting in the wings is a letter of apology for the teacher and for the student regardless as to who started it. The letter includes how my child did not use the best of judgment in fighting in the first P. because it is not acceptable behavior - but again, we have never had that problem with either of the girls.
The teachers have always seemed to appreciate the fact that the girls have had to write their letters explaining what they did wrong, why it was wrong, and include an apology for disrupting class or whatever part of school process they disrupted in addition to how this will not happen again.
All in all, it is more about making the child take responsibilities for their actions. Kindergarten is a good P. to start. As a parent, you can help as your child puts the words down in their own words as to what they did wrong, why, and how it will not be a problem in the future, then make sure the child understands this cannot happen again. It isn't so much about punishment at home as it is ensuring the child owns what he or she has done and understands why the infraction is not acceptable. I often use an example of a scenario where my child is the victim in all this to get the point across. I use explanations such as, "what if the teacher did not grade your paper and she marked it with a zero because she did not want to take the time to do what is expected of her/him? Do you believe this is right? Wouldn't you like to receive an apology from the teacher for her/him not doing their job and giving you assurances that it won't happen again? It usually works perfectly.
It is a matter of teaching the child that actions have consequences and those consequences can disrupt class or another areas of school development the child will miss out on if the child is in trouble or trying to cause trouble so it is a lose-lose case for all.
I hope this helps. It's worked for our oldest and so far, it is working for our youngest too. In addition, I write each of the teachers that work with my kids (now just the one at home of course) throughout the school year and let them know they have free reign to contact me with any problems and as such, they have my word I will make sure my child understand what they did wrong, why it is unacceptable, and how the proper thing to do is to apologize for the infraction in question.
Every teacher I have dealt with over the years has always felt comfortable with sending me an email or calling to let me know if one of the girls were straying off course. Between the both of us, we are able to get their heads back in the game because it is so important to learn what they do each day.
Hope this helps...
PS: My oldest had to write a letter to her kindergarten teacher apologizing for not doing her homework when she had the one weekend with her birth father and his family. she came back home late Sunday night (the last time they ever had her for visitation after that) but did not mention she had homework undone until I heard from the teacher. Oh I was at that school ready to lay it on her when the teacher calmed me down then showed me a note sent by my ex-husband's mother on why she did not do her homework; it was completely bogus reasoning so I had her teacher gather up extra work she was doing at home to make up for what she missed. To this very day, both girls will stay inside and off the phone even if a friend calls and out of the cats while typing up essays on the computer. (she doesn't know it but there is a safety program on her computer that not only records keystrokes but also the time stamp. if they break the rules, I take on an extra week.
Again, I hope this helps.