Add vibration to incoming Signal messages (iOS)

Look in Settings > Notifications for an app that only shows "Sounds". If you have more than one app, turn on Badges or Banners (temporarily) until you get the next notification. The app with the new badge should be the culprit.

Otherwise, go through the list one by one and disallow Sounds for each app successively until you find the offending app.

The only solution that actually worked was a full factory reset:

• Make a backup of your iPhone using iTunes.
• Using iTunes, perform a full reset of the iPhone.
• After the iPhone reboots, disconnect it from iTunes and go through the initial setup.
• Connect the iPhone back to iTunes.
• When asked by iTunes, choose to restore the backup.
• If applications are missing after the restore, perform a full sync to have them re-installed.

I also called Apple support, and they confirmed this is a known and common problem when upgrading to IOS 6, and a full reset is the only way to permanently solve it.