I supplement loads and am fed up too of having to live my day around a supplement schedule.
Supplements are a personal choice and depend on your diet, life style and deficiencies. Ideally it would be great to be tested for every vitamin and mineral and consult with a nutritionist but this isn't really practical so many choose a general assortment known to be beneficial to people with thyroid conditions.
Doctors only generally consider tests a supplements for iron and B12 to be essential as deficiencies in these are common but serious.
As a starter I would recommend a good B complex with high B12. (If deficient in B12 you will need to supplement separately aw well to raise to optimal level.)
Vitamin C at least 500mcg with bioflavonoids to ensure good absorption.
I supplement selenium because it is known for excellent thyroid hormone med conversion T4-T3.
A good quality fish oil ( Eskimo are popular) for omega oils as "thyroidys" are usually dry...as in skin, hair, eyes...
There are plenty of others and other members will advise you that this is best and that is best but you need to decide yourself.
I believe some supplementing is essential as thyroid problems can put a terrible strain on the body and cause malabsorption problems leading to deficiencies.
Other common supplements to consider are Vit A, Vit D, Vit E, zinc & magnesium, calcium...the list is endless... And of course iron if you are deficient.