**Core Concept**
The question is testing the anatomical relationship between the superior thyroid artery and surrounding nerves. The superior thyroid artery is a branch of the external carotid artery and supplies the thyroid gland. During its ligation, the surgeon must be aware of the nearby nerves to avoid injury.

**Why the Correct Answer is Right**
The external branch of the superior laryngeal nerve (EBSLN) is the most commonly injured nerve during ligation of the superior thyroid artery. This nerve arises from the superior laryngeal nerve, which is a branch of the vagus nerve, and innervates the cricothyroid muscle. The EBSLN is closely associated with the superior thyroid artery, as it runs between the artery and the thyroid gland. Injury to this nerve can lead to hoarseness or weakness of the voice.

**Why Each Wrong Option is Incorrect**
**Option A:** The recurrent laryngeal nerve is a branch of the vagus nerve that supplies the laryngeal muscles, but it is not typically injured during ligation of the superior thyroid artery, as it is located more posteriorly.
**Option B:** The internal laryngeal nerve is a branch of the superior laryngeal nerve that provides sensory innervation to the larynx, but it is not as closely associated with the superior thyroid artery as the EBSLN, making it less likely to be injured during ligation.
**Option C:** The hypoglossal nerve is a branch of the vagus nerve that supplies the tongue muscles, but it is not related to the superior thyroid artery and is not typically injured during ligation.

**Clinical Pearl / High-Yield Fact**
The EBSLN is often at risk during surgical procedures involving the thyroid gland, and its preservation is crucial to maintain normal voice function.

**Correct Answer: C. The external branch of the superior laryngeal nerve (EBSLN).**