Summary

If you haven't yet created a fix version in project admin -> versions, then it isn't possible to edit the "fix version/s" field anywhere.
I.e. you see this (e.g. on edit issue dialog, or view):

But then once you've created at least one fix version, you see this:

Steps to reproduce:

1. Create a new project (without any fix versions added yet)
2. Ensure the "fix version/s" field is on all required screens, inside the correct schemes etc (e.g. view and edit screens)
3. Ensure the current user has "resolve issue" permission (required to edit the "fix version/s" field)
4. Create a new issue and navigate to it
5. Open the edit dialog

Expected result:

The "fix version/s" field is editable, with the option to choose "new version"

Actual result:

The field is not editable, with value displayed as None

Workaround

Add the new version from Project Administration.

Notes

It is possible to create new "fix versions" from the drop-down, so this restriction shouldn't exist.