Open Settings
Tap Notifications
Tap Messages
Turn off Allow Notifications
Or tap Sounds and select None if available
Turn off Lock Screen alerts if you want no visible alerts
Turn off Notification Center alerts if you want fewer notifications
Turn off Banner alerts if you want no pop-up alerts
Open the Messages app
Swipe left on a conversation
Tap the bell icon to mute that thread
Open a conversation
Tap the contact name at the top
Turn on Hide Alerts
Use Focus mode to silence all notifications temporarily
Turn on Silent Mode with the Ring/Silent switch
Lower the ringer and alert volume in Settings or with the volume buttons