After a bit of research I finally bought a set of Beyerdynamic DT 240 Pro headphones. They are not one of the models recommended in this thread, but they are more affordable than those, and so far seem to be exactly what I need. I can hear the bass clearly when I'm transcribing a song, and they respond well to changes in the EQ when I fiddle with the settings in the MAXXAudio Pro program on the PC. (Yes, I know, that program isn't everybody's cuppa, but it comes installed on the PC and works well for my limited needs.)
The pros of the headset are: smaller than the PJBs I used to have, so more manageable for a petite person like me; they have two inputs for the cable, one on each earpad, so if one goes noisy or fails, you have a second option; they are tighter than bigger models, so they don't fall off my little head when I stand up to play bass... 😉
So far I've found only one con: the cable they come with is too short, despite having a curly section that you can stretch if necessary. I have ordered a 3m cable with standard 3.5mm to 2.5mm male inputs, and hopefully that will solve the issue.