Hi Schwaben10,
If you are not always seeing text message notifications on your home screen, take a look at the resource below. I've highlighted some key points here, but you can click on the link for complete information.
About Notifications on iPhone, iPad, and iPod touch - Apple Support
https://support.apple.com/en-us/HT201925
To see which apps provide notifications and change notification settings, go to Settings > Notifications, and tap the app name (in this case, Messages).
- If you want the notification to appear on your screen, set the notification type as an Alert or Banner.
-
You can access your Alerts and Banners Notifications from any screen, including the Lock screen. Just swipe down from the top of the screen. You can also tap a notification in the Lock screen to open the related app.
-
By default, your Notifications will show you the Today view, which gives you a summary of what you need to know for the day and a glance at tomorrow. The Notifications view lets you see all of your alerts or those you haven't addressed in the last 24 hours. To see the Notifications view, swipe left while on the Today view.
Push notifications require an active Wi-Fi or cellular connection. If you’re not getting notifications for a specific app, try these steps:
- Go to Settings > Notifications to verify that the app supports notifications. If notifications do not appear in the Notification Center, make sure that the Notification Center setting for the app is enabled.
- Make sure you’re signed in to your Apple ID.
Notifications use Wi-Fi only when a cellular connection is unavailable. Firewalls and proxy servers may affect your ability to receive notifications. Get help using Apple PushNotifcationservice (APNs).
Schwaben10, if the notifications issue is only being seen with one or a few contacts, you can try to remove the contact, restart your iPhone, add them again and then reset the notification preferences.
Regards,