In the basic search Severity will appear in the list of custom fields. I'm not sure what you mean by "displayed between other fields", as it will be be on the list of custom fields.
On advanced search, obviously, you just type the name.
If your field does not appear in basic's list of fields, or gives you a "field not known" when naming it in an advanced search, then the most likely reason is that you have not given the field a searcher.
New fields always have a default searcher, but admins can remove them. Go to Admin -> issues -> Custom fields, and find "Severity". Click "Edit" (not "configure") and you'll be offered edit boxes for the field name, description and searcher. Make sure severity has an appropriate searcher set. If you need to change it, you will need to re-index after changing it for searches to work properly.
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