**Core Concept**
The question requires knowledge of the anatomical relationship between the thyroid gland and surrounding nerves, specifically the recurrent laryngeal nerve (RLN), which is at risk during thyroid surgery. The RLN is a branch of the vagus nerve (CN X) that supplies the laryngeal muscles and is closely associated with the thyroid gland.

**Why the Correct Answer is Right**
The recurrent laryngeal nerve is the most common injured nerve after thyroidectomy due to its close association with the thyroid gland and the surgical field. The RLN courses anterior to the pretracheal fascia and lateral to the thyroid gland, making it susceptible to injury during thyroid surgery. The risk of injury is increased by the use of retractors that elevate the thyroid gland, which can stretch or compress the RLN.

**Why Each Wrong Option is Incorrect**
**Option A:** The external branch of the superior laryngeal nerve (EBSLN) is another nerve at risk during thyroid surgery, but it is not the most common injured nerve. The EBSLN is responsible for innervating the cricothyroid muscle, which is involved in vocal cord tension.
**Option B:** The hypoglossal nerve (CN XII) is not typically at risk during thyroid surgery, as it is located more laterally and is not as closely associated with the thyroid gland.
**Option C:** The vagus nerve (CN X) trunk is not typically injured during thyroidectomy, as the RLN is a branch that is more susceptible to injury.

**Clinical Pearl / High-Yield Fact**
During thyroid surgery, the recurrent laryngeal nerve is at risk of injury, and the use of intraoperative neuromonitoring can help identify and protect the nerve.

**Correct Answer: C. Recurrent laryngeal nerve.**