What tests/imaging technique shows nerve avulsion/detachment/tear? can they be repaired?

Yes: Mri and yes they can be repaired -- go to ucla center of excellence for our southwest region :).

MRI: If you are looking for soft tissue injuries, an MRI is often the best test. Nerve damage is often temporary and the nerves regrow / repair in 6-12 weeks. Muscle/tendon tears often need surgery.

Depends on where: But the bigger question may not be whether the nerve is torn (it would be obvious if it is - you'd have paralysis, pain, and other clinical signs), but whether the nerve if functional. It would not be functional in case it was torn but also for other reasons. If you want to assess whether a nerve is functional or not, and EMG with a nerve conduction study (ncs) is the tests of choice.