Anchor the answer to vertebral levels of the neck, because each midline cartilage maps to a spinal segment. The hyoid bone sits at C3, the upper edge of the thyroid cartilage at C4, and the cricoid cartilage at C6.
The common carotid artery climbs the neck and splits into internal and external branches exactly at the C4 level. Since the upper border of the thyroid cartilage is the palpable marker for C4, that is where a clinician feels the bifurcation and the carotid sinus. The cricoid at C6 is too low and corresponds instead to where the pharynx becomes oesophagus and larynx becomes trachea. The hyoid at C3 is too high. The cricothyroid membrane is an airway landmark used for emergency access, unrelated to the arterial split. Matching the artery to its C4 level leaves only the upper border of the thyroid cartilage.
\[\boxed{\text{Upper border of thyroid cartilage}}\]