When a doctor requests an mri what is the criteria for requesting this test with or without contrast? if a tear is suspected, is contrast best?

It depends: Contrast injected in a joint can increase the likelihood of identifying a subtle abnormality (e.g., labrum injury, post-op knee) on mri. In many cases, a ligament injury will be visible without the contrast. Factors determining whether contrast is useful (or necessary) include which joint is being imaged, what the suspected injury is and the personal experience/training of the doctors.

Often without: When looking for a muscle, ligament or tendon tear, contrast is not needed. Contrast is more often used when looking for diseases of the solid organs (liver, kidneys, spleen, etc) or evaluating blood vessels or evaluating tumors. It would be impossible to go over all of the criteria, but every MRI request is reviewed before the test is done to make sure it can evaluate the suspected problem.