Most common type of carcinoma of thyroid
Papillary carcinoma
Papillary carcinoma is the most common thyroid malignancy. 
Nonetheless, when papillary cancer is diagnosed most patients will be  offered  treatment.  The  disease  is  known  for  its  propensity for lymph node metastases. These are more common in younger patients, in whom they do not affect the otherwise excellent survival.
Reference: Bailey & Love’s short practice of surgery , 27th  ed page no 818
Papillary
Follicular
Medullary
Anaplastic