  We prove that the geodesics associated with any metric generated from
Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are
necessarily singular with respect to the law of any type of SLE$_\kappa$. These
hypotheses are satisfied by the LQG metric for $\gamma=\sqrt{8/3}$ constructed
by the first author and Sheffield, and subsequent work by Gwynne and the first
author has shown that there is a unique metric which satisfies these hypotheses
for each $\gamma \in (0,2)$. As a consequence of our analysis, we also
establish certain regularity properties of LQG geodesics which imply, among
other things, that they are conformally removable.
