On Wink, you cannot delete individual messages. However, you can delete whole discussions by unfriending someone or deleting a chat. When you unfriend someone, you will be removed from their match list, as well as from yours.
To unfriend someone or remove a chat, go to your match's profile and click the menu icon (ellipses) in the upper right corner, then select Unfriend or Delete Chat.
Note: Unfriending is a permanent action.