I am applying for a new job, but do not want to let my current supervisor know I am looking elsewhere. However I want to put him as a reference because I know he will say positive things about me to my possible new employer. Do you think they will call him before I interview or after?
Before the interview, say "references available on request" in any application form that asks for you to provide references when you apply. Do not write them on your resume. Do not include the name or contact information of your supervisor on your resume.
After the interview, if they want references, remind them that you are still employed at [CurrentEmployer] and have not yet given your notice. Ideally, a reference check would be one of the final steps in this process.