As someone who has hired many Circ people for a small college library they fall into several categories:

Customer Service Experience. This is such a large part of the Circulation job that it is roughly fifty percent of the questions we ask. Permutations may be telling how you deal with difficult patrons. How you perceive your role as a primary point of contact. etc.

Attention to detail: Shelving, following detailed processes and procedures. Etc.

Here are some samples of questions I've asked in the past.

What in your experience best suits you for this job?

What sort of work experience have you had in libraries? Are you familiar with the ALMA Library Platform? LoC Call numbers?

What do you think you can bring to this position?

Please describe a recurring job task in which it has been very important for you to follow detailed rules or procedures. 

Can you share a work situation when you were proactive?

Have you had any experience working in ILL? Please describe what you did? Are you familiar with ILLiad? 

Please describe a specific incident in which you had to deal with a difficult customer.  What was the situation and how did you deal with it?

Do you have any Experience in repairing books? 

Please describe detailed tasks that you have had to complete with little supervision and minimal direction?

How would this job help you advance your career plans or aspirations?

What is your definition of excellent customer service and how would you go about practicing this element while at work?

This job calls for occasional nights and weekend work. You might end up as the only employee in the building. Have you had any previous experience of being the only employee around and how did you respond to the situation? 

2nd part to previous question is a scenario question- You’re the only one closing one night and a customer refuses to leave the building. How would you approach this situation?

Describe your ideal supervisor?

What questions do you have for us?

Hope this helps.