I guess the most typical would be that $P(X_t=x)=0$.

Say, if $Y$ is a Brownian motion, $Z$ is an independent pure jump process and $X$ is a jump diffusion given by $X=Y+Z$, then for $t>0$, $$ P(X_t=x)=P(Y_t=x-Z_t)=0 $$ since $Y$ is independent of $Z$.

As long as $\sigma\ne 0$, we get $P(X_t=x)=0$.

By the Levy-Ito decomposition there is a Brownian motion $Y$ and an independent process $Z$ with $X=Y+Z$. Then for $t>0$, $$ P(X_t=x)=P(Y_t=x-Z_t)=0 $$ since $Y$ is independent of $Z$.