I will report on recent investigations, in part joint with A. Martini (Politecnico di Torino), on global regularity properties of sum-of-squares operators that fail to satisfy the celebrated Hörmander condition on a "thin" set. Under appropriate assumptions on the structure of the thin set in question, the operator is shown to be globally hypoelliptic despite not being locally hypoelliptic, and sometimes even to satisfy a stronger "Kohn-Nirenberg compactness estimate".