Open Snapchat
Tap your profile icon
Tap the gear icon to open Settings
Scroll to Privacy Controls
Tap Clear Conversations
Tap the X next to the chat you want to clear
Confirm if prompted
Repeat for any other chats you want to remove from the chat list
To delete a specific message, open the chat, press and hold the message, then tap Delete
To clear all saved messages, open each chat and delete saved messages manually