Do Navy divers get to serve and deploy on aircraft carriers? Or strictly on smaller vessels and near broken wreckage?

You don't want to be on a carrier anyway. Carrier life sucks. Everyone below the rank of O5 is basically a peon, lines are terrible for everything (chow, ship's store, liberty, you name it), the whole ship is unionized into the different shops (rather than being a one-team-one-fight cohesive crew), you don't get good liberty ports very often, and when you do get to go somewhere other than Bahrain or Jeb, you have to anchor out most of the time... try to run liberty boats for 5000 people... you end up standing in line for hours waiting on a launch.

Small ship life is the way to go.

Hi Max. Chief has it right. Specialists are assigned to units that use divers to execute their mission. Carriers don't have a diving mission along with most other classes of vessels. For any production vs. support skillset, just look for who/what has that production mission. We do have divers in the Seabee Underwater Construction Teams and also divers in the Explosive Ordnance Disposal units. SEALS obviously dive. So divers are around, just not that many of them.

Max Mitchell The only vessels divers are assigned to are the Sub Tenders now as part of the repair crew. My first ship in the Navy however, was a rescue salvage ship, the USS Grasp ARS-51 on which we had divers assigned as ships crew. Those rescue salvage ships were transferred to the Military Sealift Command, and have a civilian crew now. I think they only embark a detachment of divers as needed.