In this talk, we prove an upper bound on the average number of 2-torsion elements in the class group of monogenised fields of any degree n≥3 and, conditional on a widely expected tail estimate, compute this average exactly. As an application, we show the existence of infinitely many number fields with odd class number in almost every even degree and signature. Time permitting, we will also discuss extensions of these results to orders (joint with Shankar, Swaminathan and Varma) and the relative setting.