Settings > LED Indicator > Make sure "Missed Event" is checked.
Go under "settings", then "sounds", then under "ringtone and notifications" look for "Default Notifications" and make sure that you have a sound selected and that it is not set to silent.
It's not that I'm not getting sounds or LED notifications, I just don't have any voicemail notification at all--no little tape at the top indicating that there is a voicemail.
I can't always get to my phone so I rely on my voicemail notifications, too! I completely understand the importance of having that icon pop up for you! Although, mrsdawg2 and maya0630 both provided some helpful information, it looks like you've already taken these great troubleshooting steps. I would like to recommend resetting the Voicemail application on your Galaxy S III:
• From the home screen, select Apps (located in the lower-right).
• From the APPS tab, select Settings.
• From the DEVICE section, select Application manager.
• From the All tab, select Voicemail.
• Select Force stop.
• Select OK.
• From the STORAGE section, select Clear data.
• Select OK.
Please test your voicemail again, after these steps. Let us know if you need anything further.
Follow us on Twitter@VZWSupport
Also you may want to call and check your advance settings on the voicemail system, there is a option to disable notifiaction located there also..
Call Voicemail from phone and login, enter optiions menu and check for voicemail notification (Or Something close to that) because I am using YouMail and cant remeber the correct menus.