  In this work, we establish a comparison principle for stochastic Volterra
equations with respect to the initial condition and the drift $b$ applicable to
a wide class of Volterra kernels and driving forces $g$ that may be singular in
zero. For completely monotone Volterra kernels such a result holds without any
further restrictions, while for not completely monotone kernels it is shown
that such a principle fails unless the drift is additionally monotone. As a
side-product of our results, we also complement the literature on the weak
existence of continuous nonnegative solutions. This covers the rough
Cox-Ingersoll-Ross process with singular initial conditions.
