Supracondylar humeral fractures are the classic childhood elbow injury, and a notable share of the displaced ones carry a nerve lesion, so identifying the commonest is high-yield.
An important point is that the answer has changed with evidence. Earlier teaching pointed to the radial nerve, but more recent series show that the anterior interosseous nerve, the pure motor branch arising from the median nerve, is the one most often affected. Its involvement is tied to the usual posterolateral displacement of the distal fragment seen in extension-type fractures.
The displacement direction predicts the nerve at risk: posterolateral displacement threatens the anterior interosseous and median nerves, posteromedial displacement endangers the radial nerve, and the uncommon flexion-type fracture puts the ulnar nerve at risk. Clinically an anterior interosseous palsy causes loss of the OK pinch from weak flexor pollicis longus and index profundus, characteristically without sensory loss.
Hence the nerve injured most frequently is the anterior interosseous nerve.
\[\boxed{\text{Anterior interosseous nerve}}\]