  Every pseudo-Anosov mapping class $\varphi$ defines an associated veering
triangulation $\tau_\varphi$ of a punctured mapping torus. We show that
generically, $\tau_\varphi$ is not geometric. Here, the word "generic" can be
taken either with respect to random walks in mapping class groups or with
respect to counting geodesics in moduli space. Tools in the proof include
Teichm\"uller theory, the Ending Lamination Theorem, study of the Thurston
norm, and rigorous computation.
