This is one of the common issue those will be addressed by Apple very soon with their iOS10 updates, the issue appeared to be no alarm sound, no screen on or icon change.
Apple introduced Bedtime feature in iOS10 and most of the people who upgraded their phones to the latest iOS are the one who have seen the issue so far.
Apple hopefully will address this issue soon and we’ll update here if that ever happend.
The only solution exists for this is wipe the phone and re-install the iOS10. I know it is a pain and nobody wants to do that, but apparently it appeared to fix the issue.
If you don’t want to do that you can keep using the Bedtime feature on iOS10 to set alarms.