I posted some of the varied solutions here:
Text I posted is here:
I've been reading and researching this the last several hours on various threads, and users have complaints from all carriers (AT&T, Sprint, Verizon, T-mobile) and even in some cases past issues with different iOS and iPhones... one thing is consistent - it is theiPhone. So let's move on.
To all, several suggestions are reported with varying success amongst users:
A) Delete 1 voicemail. Scroll down to bottom, select Deleted Messages, Select Clear All for the 1 voicemail. Repeat. (basically delete then delete, one at a time. This one worked for me!)
A1) If the "Clear All" is greyed out at first, select then deselect the message, and "Clear All" should be blue and selectable.
B) Call AT&T (or whoever your carrier is) and request them to "reset your visual voicemail" or a "voicemail system reset". Reference this help thread if you need to (see note by Brucku). Solved for 2 other people. The thread is here:
C) Call your own voicemail manually (call 1 or call your own #) (circa early 90s) and delete each message manually through the voice prompts (press 7 to delete, for AT&T at least). You may need to make sure to "listen to saved messages" as some are lurking in there. Reportedly solved for quite a few, but not working for many others.
D) Something convoluted about deleting the Visual Voicemail message, then Clear All, then enter Airplane mode, restart the phone (home + power button), then turn off Airplane mode. Few said worked, but most said it didn't. (I tried several variations of this to no effect).
E) Reset "Network Settings". (note: this will reset wifi passwords, but phone is fine just don't Reset All!) Reboot phone. Delete messages as you would. Worked for some.
F) Drop iPhone in toilet. All Visual Voicemails should disappear. Works for all who are willing to part with phone.
Hope this is helpful. So far sol'n "A" is working for me, and this was affecting 2 phones in my household. Will update if it reverts and I have to do "F". Cheers!!