You can archive your messages in the app. When messages are archived, they do not appear in the list of all messages.
You can archive a message by swiping it to the right.
Example on Android
You can archive the message by swiping it to the left and select Archive.
Unarchiving your message
If you want to move your message back from the archive to the list of all messages, you can click on the All messages dropdown in the menu.
Click Archived to see all archived messages.
Swipe the message again to unarchive the message.
To view the list of all messages, including your unarchives message, you can click the dropdown button in the menu again, and select All messages.
You can not permanently delete your messages. All messages are securely saved on the servers, and are synchronised with the messages in the web application.
You can however delete the locally stored data from your phone, such as your login code, through your Settings.