**Core Concept**
Supracondylar fractures of the humerus are a common type of elbow fracture in children, typically occurring in the distal humerus above the condyles. These fractures often result from a fall onto an outstretched hand, leading to displacement of the distal fragment.

**Why the Correct Answer is Right**
The correct answer is related to the common complications associated with supracondylar fractures. The most significant concern is the potential for neurovascular injury, particularly to the brachial artery and median nerve. This can lead to ischemia of the forearm and hand, necessitating prompt recognition and management. The displacement of the fracture can also lead to compartment syndrome, which requires urgent decompression.

**Why Each Wrong Option is Incorrect**
**Option A:** This is a true statement about supracondylar fractures. They are indeed most common in children between the ages of 5 and 7 years.

**Option B:** This is a true statement about supracondylar fractures. The fracture is typically classified based on the degree of displacement, with type I being minimally displaced, type II being posteriorly displaced, and type III being completely displaced.

**Option C:** This is a true statement about supracondylar fractures. The most common complication is indeed neurovascular injury, particularly to the brachial artery and median nerve.

**Option D:** This is the incorrect statement about supracondylar fractures. The most common complication is not fat embolism syndrome, but rather neurovascular injury and compartment syndrome.

**Clinical Pearl / High-Yield Fact**
A key clinical pearl to remember is that children with a supracondylar fracture often present with a characteristic "popeye" deformity due to the displacement of the fracture. This is a critical sign to recognize, as it can indicate the need for urgent surgical intervention.

**Correct Answer: D. Fat embolism syndrome is not the most common complication of supracondylar fractures.**