Following a method introduced by Thomas-Vasquez and developed by Grundman, we prove that many Hilbert modular threefolds of geometric genus 0 and 1 are of general type, and that some are of nonnegative Kodaira dimension. The new ingredient is a detailed study of the geometry and combinatorics of totally positive integral elements of a fractional ideal in a totally real number field that are minimal with respect to trace up to multiplication by totally positive units.