Hide message previews on the Lock Screen: Settings → Notifications → Messages → turn off “Show Previews”
Hide message notifications entirely: Settings → Notifications → Messages → turn off “Allow Notifications”
Hide message alerts for a specific conversation: Open Messages → tap the contact/thread → tap the contact name → turn on “Hide Alerts”
Hide message content by filtering message requests: Settings → Messages → turn on “Filter Unknown Senders”
Hide conversation threads from the main list: Open Messages → tap the Filters/Unknown tab (if enabled) and use filtering options
Hide SMS/iMessage app badge: Settings → Notifications → Messages → turn off “Badge App Icon”