MSN Live Messenger allows you to log all your conversations in a folder on your computer. This setting can be found under Tools > Options > Messages > Message History.
To enable or disable conversations logging, just check or uncheck the tickbox that says “Automatically keep a history of my conversations”. You will also be able to choose where your log files will be stored into. The default folder will usually be C:\documents and Settings\Username\My Documents\My Received Files\History.
To delete chat log, just browse to that folder and search for files ending with .xml. You can delete all the chat logs if you want or you can choose to delete only the chat history that you had with a specific person. Say for example, that you want to delete your conversations history with Jack, just delete jackxxxx.xml and that will do.