If $M$ is smooth and $\dim M =3$, then the answer is always. This is a theorem due to Shokurov, see
V.V. Shokurov: Smoothness of a general anticanonical divisor on a Fano variety, Math. USSR, Izv 14, 395-405 (1980).
In fact, Shokurov proves the following more precise result.
Theorem. Let $M$ be a smooth Fano threefold of index $r$ and let $H \in \textrm{Pic}(M)$ such that $rH \cong -K_M$. Then the general element of the linear system $|H|$ is smooth.
Using a terminology due (I think) to M. Reid, smooth anticanonical divisors in Fano threefolds are sometimes called elephants.