Open the Messages app
Swipe down on the conversation list to reveal the search bar
Tap the search bar at the top
Type a name, phone number, keyword, or phrase
Review the matching messages and conversations
Tap a result to open it
In a conversation, swipe down to search within that thread
Use Spotlight Search by swiping down on the Home Screen and typing a message-related term
Check the search results under Messages
Search in Mail, Notes, or other apps if the message content was copied there