Why can't I delete an issue?

That's a question that involves various levels of configuration in JIRA, but in this case has a nice, simple answer.

What does deleting a JIRA issue mean?

Deleting an issue in JIRA is permanent, making it almost as though the issue had never existed. This is usually only useful when you've created an issue by mistake or have been creating issues to test JIRA itself. After a deletion the only way you can usually tell that there used to be an issue is that there is a gap left in the issue keys, e.g. TEST-12, TEST-13, (missing issue), TEST-15. The missing issue TEST-14 may have been moved to another project, in which case JIRA will redirect to the new issue key. But if there is no helpful redirection then the issue was probably deleted.

Who can delete an issue?

The permission to delete an issue is named "Delete" in a permission scheme. For a given project check which permission scheme is being used and who has the Delete permission in that scheme. By default the permission is granted to a Project Role such as Administrators. Then check which user has that role for the project.

But I still can't delete any issues!

It's time to look deeper into the bowels of JIRA. Deleting an issue is a JIRA system issue operation, just like Linking or Moving an issue. These operations are controlled by a system plugin unsurprisingly named "Issue Operations Plugin". If you go to Admin, System, Plugins then it's there in the ridiculously long list of plugins. The Universal Plugin Manager hides these plugins by default. If an issue operation is disabled in the Issue Operations Plugin then it won't appear anywhere in JIRA at all. 

Summary: if you can't delete any issue in any project, it's probably because deleting has been disabled in the Issue Operations plugin.

Extra tip: new issue operations can be added to JIRA using a plugin and the Issue Operations Plugin Module.