Appointment booking is probably the most obvious clusterfuck. I still can't book appointments with my GP online.
We got a referral to ENT for a child. They sent us a letter, telling us to call them. On the phone they said they have chosen an arbitrary appointment date for us and will send the details in another letter. When we get that letter, if the time isn't convenient then to call them back to rearrange it. Madness!
It is very very slowly improving, so I think eventually these sorts of brain dead issues will mostly go away. Hopefully while I'm still alive.