In a good regulator design you will see around 1 mv of noise, that's a bit more than -60 db down, not bad. ...
In as this is for a DC heater design, most of the audio concerns can be eliminated.
Some tube microphones use the DC heater for grid biasing too.
In this case additional filtering by a large cap is desirable.
Even if this is not the case, noise current introduced into the common ground connection might creep into the audio path.
A 4700uF / 16V cap doesn't cost a fortune.