Which nerve extends to the muscles on the side of the neck?

The cervical cutaneous nerve is the correct choice because it specifically innervates the muscles and skin on the side of the neck. This nerve is part of the cervical plexus and plays a crucial role in both sensory and motor functions in that region. Its branches are responsible for providing sensation to the cervical skin and facilitating movement in the neck muscles.

The other nerves listed do not primarily serve the muscles on the side of the neck. The trigeminal nerve primarily carries sensory information from the face and controls muscles used for chewing. The facial nerve is mainly responsible for controlling the muscles of facial expression and has limited involvement in neck muscle innervation. The vagus nerve has a wide range of functions, including autonomic control over the heart and digestive tract, but does not specifically target the muscles on the side of the neck.