Rarest type of thyroid carcinoma is
Undifferentiated (anaplastic) carcinoma
This is one of the most aggressive malignancies in humans. 
Thankfully it is rare. It may develop de novo, or present as dedifferentiation of a papillary or poorly differentiated carcinoma. The disease is characterised by rapid growth, visceral invasion  and  distant  metastases. The  surgeon’s  role  in  this disease is crucial. Thyroid lymphoma can be incorrectly diagnosed as anaplastic cancer and so biopsy is critical. This can be done using a core or open technique.
Reference: Bailey & Love’s short practice of surgery , 27th  ed page no 816
Papillary
Follicular
Medullary
Anaplastic