For example, I know that the non-regular language $a^nb^n$ is in $AC^0$. I would like to know more examples like this.
Languages in $AC^0$ can be more complicated than naive intuition might suggest.
$AC^0$ is closed under logical operations, concatenation and composition, so you can combine above examples. Now you should feel some respect for $Parity \notin AC^0$ and other circuit lower bounds!