What do you put on a job application as reason for leaving when you should have been promoted and someone else was or a bad work environment?

You need to be careful here not to be dishonest, but at the same time, you need to clarify or explain your story. First of all, if you left a position because of a bad work environment, stick to that story. Talking about someone else getting promoted and you were not promoted shows that your ability to handle situations with people is not very good. If you talk about in an interview, you may tend to be negative about the situation, or you may tend to say something or write something down that may turn the next employer off from hiring you. They could get the impression that you were the actual cause of the problem and were not promoted because you were the issue, not the one given the role.

Since you mentioned a bad work environment, stick to that story and keep it brief. If you do write it down, it may come up in an interview. You may also be better off turning it into a positive thing. So for example, you could talk about a bad work environment, and then turn it positive by talking about the things that you want to have and what you would consider to be a "good" work environment. This can be a good way to lead into specific questions about the company for which you are interviewing where you can find out what it is like to work there.

The fact that you should have been promoted is a "he said, she said" type situation, so even if it were true, it will not help your situation.