Ok, there's a few more options here...
Script runner won't do it, unless what you want to do is put a normal date field on edit screens and then write a scripted field that displays the MMM-YY format you want on view (and search)
Another option, if you don't have too huge a range of dates to worry about might be a cascading select field with yyyy for the first option and then the months in the second option
Possibly the easiest option - have two fields, one with a list of years, and a second with the 12 months in it. No coding or fuss required, but they will be independent of each other (again though, a similar trick to the first one - write a scripted field that combines them for display and search)
You can use
@Boris Georgiev [Botron],
first of all thanks for your help.
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