**Core Concept**
The abductor pollicis brevis muscle is one of the thenar muscles of the thumb, responsible for thumb abduction. It plays a crucial role in fine motor movements of the hand, particularly in opposition and circumduction of the thumb.

**Why the Correct Answer is Right**
The abductor pollicis brevis muscle is innervated by the median nerve, specifically the recurrent branch of the median nerve. This branch arises from the median nerve as it passes through the carpal tunnel and supplies the thenar muscles, including the abductor pollicis brevis. The recurrent branch of the median nerve is responsible for motor innervation of the abductor pollicis brevis, allowing for precise thumb movements.

**Why Each Wrong Option is Incorrect**
**Option A:** The radial nerve is not responsible for the innervation of the abductor pollicis brevis muscle. The radial nerve primarily supplies the extensor muscles of the wrist and fingers.
**Option B:** The ulnar nerve is not involved in the innervation of the abductor pollicis brevis muscle. The ulnar nerve supplies the intrinsic muscles of the hand, except for the thenar muscles.
**Option C:** The musculocutaneous nerve is not responsible for the innervation of the abductor pollicis brevis muscle. The musculocutaneous nerve primarily supplies the flexor muscles of the arm.

**Clinical Pearl / High-Yield Fact**
The recurrent branch of the median nerve is a critical structure in the carpal tunnel, and damage to this branch can result in weakness or paralysis of the thenar muscles, leading to difficulty with thumb opposition and other fine motor movements.

**Correct Answer: A. Median nerve**