What is the correct version?
- sich bewerben um etwas
- sich bewerben für etwas
Are there cases where one or the other is preferable? Please make some examples if there are.
To not only comment but finally answer the question, there is quite a concise list of examples in Duden
They all show the usage of "um" as the preferred preposition used. Still, Duden says that it is not wrong to use "auf". Anything else, especially "für" should not be used, even less so if it is in a letter of application. You may hear it in colloquial speech though.
This question is already answered by Takkat, but i would like to add some Google NGram comparisons between the differen prepositions in connection with "bewerben":
At a first glance it seems that using "für" is a quite new trend, having gained a considerable currency even in written language. But there are cases of using "für" where other prepositions can't be used:
and so on. So, the counts for "auf" may indicate in such phrasing, not necessarily in "sich für einen Posten bewerben".