Text Notifications

An additional thing to check is the mute switch over the volume buttons.

You can set a Text Tone here:

Settings > Sounds & Haptics > Text Tone

Also check if Hide Alerts have been activated for one or more of your SMS conversations. Look for a small "crossed out bell". out of the conversations. You deactivate Hide Alerts by swiping from left to right on the conversation in the overview and tapping the purple bell.