Greetings, mah02.
It appears your iPhone 5s on iOS 10.2 isn't notifying you when you receive information from many applications. You'll see them once you open the app, but not before. I depend on notifications myself and I'll be glad to help.
To start, force close all of the apps that are open on your iPhone, then restart:
Force an app to close on your iPhone, iPad, or iPod touch
Restart your iPhone, iPad, or iPod touch
Once your phone has returned to the home screen, send yourself an email from your email account on a browser. If you still aren't getting a notification, continue below.
Use Notifications on your iPhone, iPad and iPod touch
To get notifications, connect to a Wi-Fi or cellular network. If you don't get notifications for a specific app, try these steps:
- Make sure that the app supports notifications. Go to Settings > Notifications, select the app, and make sure that Notifications are turned on.
- If you have notifications turned on for an app but you're not receiving alerts, the alert style might be set to None. Go to Settings > Notifications and check that your Alert Style is set to Banners or Alerts.
- Make sure that you’re signed in to your Apple ID. Go to Settings > iTunes & App Stores and enter your Apple ID and password.
- Make sure that Do Not Disturb is turned off. Go to Settings > Do Not Disturb and tap Manual if it's turned on.
- If you’ve recently installed an app or restored from a backup, open the app to start getting notifications.
Let me know how that goes and thanks for using Apple Support Communities!