Is the software on your phone up to date? Go to Settings > Software update > Download and install. Also, let's try a soft reset on the phone. This will end all processes and give the phone a fresh start. Press and hold the Volume Down and Power keys at the same time until the phone switches off and back on again.
Let us know how you get on!
Do you have the 'Do not disturb' feature switched on? Pull down the Quick panel menu from the top of your phone to see if this is active.
@Banno: If you head to Settings > Apps > Messages > Show Notifications > On, and check your options in General Notifications and New Messages, is it set to 'Alert' and 'Show As Pop-up', with a Sound assigned?
@Banno: It's great to hear that you were able to resolve your issue. The Messages app does have a limit to how many messages it can store at one time, therefore please follow the steps below to automatically delete old messages to make room for new ones.
Open the Messages app > Tap the 3 dots in the top right > Settings > More Settings > Delete Old Messages > Toggle this option on.