Which nerve is used for monitoring anesthesia during surgery?
Ref. Oxford textbook of anesthesia. Page. 449
 
Ulnar nerve is monitored because ulnar nerve supplies adductor pollicis muscle which is monitored for neuromuscular blockade during anaesthesia
 
Nerve Injuries in anesthesia
In the supine position the main nerves at risk are:

Brachial plexus (excessive abduction and extension of the shoulder, especially if the head is turned away from the abducted arm),
Ulnar nerve (direct compression in ulnar groove, especially if the hand is supinated, or stretching by acute elbow flexion)
Radial nerve (arm allowed to hang over the side of the operating table)
Supraorbital (compression by the facemask)
Radial
Ulnar
Facial
Median