Force close the apps where you don't get the badge icons.
Restart your iphone.
Reset it: press and hold Home and Power button until the Apple logo appears. Release both. The iPhone will power up on it's own. This won't remove data or change settings, so that's a good reason to try it as well.
If this doesn't help with your problems, reset all settings in Settings - General - Reset. You'll have to go through the Settings app afterwards to select everything you've set up before once more.
Your last option would be to backup your device and restore it using iTunes.