**Core Concept**
Supracondylar fractures are a type of elbow fracture that occurs above the condyles of the humerus, which are the bony prominences at the distal end of the humerus. These fractures are common in children and can have various complications due to their proximity to important neurovascular structures.

**Why the Correct Answer is Right**
The most common complication of supracondylar fractures is **median nerve injury**. This can occur due to the fracture itself or as a result of swelling and hematoma formation in the region. The median nerve is a critical nerve that supplies sensation to the hand, and injury to this nerve can result in numbness, tingling, and weakness of the hand. The nerve is particularly vulnerable in this region due to its proximity to the fracture site and the fact that it runs close to the surface of the bone.

**Why Each Wrong Option is Incorrect**
**Option A:** **Radial nerve injury** is not a common complication of supracondylar fractures. While the radial nerve is also located in the region, it is not as prone to injury as the median nerve due to its deeper location and more protected course.

**Option B:** **Fracture of the radial head** is not a direct complication of supracondylar fractures. While the two fractures can occur together, they are distinct entities with different mechanisms of injury and treatment approaches.

**Option C:** **Brachial artery injury** is a rare but serious complication of supracondylar fractures. However, it is not as common as median nerve injury and requires prompt recognition and treatment to prevent long-term sequelae.

**Option D:** **Cubital tunnel syndrome** is a condition that affects the ulnar nerve and is not a direct complication of supracondylar fractures.

**Clinical Pearl / High-Yield Fact**
Median nerve injury is a common complication of supracondylar fractures, and prompt recognition and treatment are essential to prevent long-term sequelae. A high index of suspicion is necessary, and electrodiagnostic studies may be required to confirm the diagnosis.

**Correct Answer: C. Brachial artery injury is a rare but serious complication of supracondylar fractures.**