First of all, it's important to understand that there is a difference between asking for references from a previous supervisor and just asking for references in general.
In today's litigious society, most companies are very wary about providing too much information about previous employees, beyond job titles and dates of employment. They're afraid of being sued. Some companies will also respond to the question, if asked, about if they would rehire that individual. Beyond that, however, many companies will not allow supervisors to give out employee information or references.
That can hurt your chances of being hired -- which is why you may want to provide your own references. That will allow you to only choose to list individuals who will give you a good reference -- and, more importantly, will be willing to go beyond providing "name, rank, and serial number" and give specific information about your performance and why you'd be an asset to the company.
Here are some other tips when choosing references:
- Always get the person's permission to list them as a reference.
- Notify the references when you've had an interview so they can expect the call. Let them know the name of the company and the title of the position you applied for.