General occurrence:
* Occur
* Take place
* Come about
* Transpire
* Materialize
* Arise
* Eventuate
* Behave
Unexpected or unplanned:
* Occur
* Arise
* Come up
* Crop up
* Develop
* Break out
Resulting from an action:
* Result
* Follow
* Ensue
* Stem
* Derive
Positive connotations:
* Fortune
* Luck
* Success
Negative connotations:
* Mishap
* Disaster
* Catastrophe
Informal:
* Go down
* Turn out
* Come off
The best synonym for "happen" will depend on the specific context and the nuance you want to convey.