What dog food would you recomend for a shinier, healthier coat?

Science Diet is a crappy food in my opinion.  Try to go grain free.  Add a raw egg every morning.  Get the dog dewormed.  Worms & internal parasites can cause this.  Add some fish oil to his diet.  I would try deworming first & switch to a better dog food.

Pay a nutritionist for a fresh food recipe. That way you cover your bases.

If you mean kibble, there is no one BEST food, for all dogs.  I have NEVER seen what you are feeding, as *highly recommended* by any dog food analysis/comparison sites.  Usually in dry kibble - you need a highly quality food with MEAT as the first ingredient and hopefully 2-3 meats in the first 5 ingredients.  I think a fish based food is best, in DRY food.  But as many others have said the best diet is the RAW (aka BARF - Bones And other Raw Foods) diet.  You can make your own (once you LEARN HOW) and totally control all sources and ingredients or buy pre-made.

Raw/bone or prey model and grooming...... ensures they get the right healthy natural food and you brush out any dead hair and dander ensuring their natural coat oils keep their coat in top shape

Fresh caught raw obviously

Try cod liver oil added to the food.