I think we have cracked the modem problem. As a result of attempts to isolate various programs, the frequent dial up message was replaced by an equally frequent message asking what connection I wanted to use, viz.
NETWORK CONNECTIONS !!! Showing
The use of the word "Network "in the title bar led us to scrutinise Internet Explorer and Outlook Express once again and then to look at MSN Messenger for the first time. We then unchecked msmsgs in System Configuration/Start Up and the frequent attempts to redail, or to ask for a connection, stopped.
We then tried to locate the sub program within Messenger that was causing the problem, but could not find it. So we removed the entire Messenger folder from c:\program files and I have had no trouble since. Shan't re-install MSM for a while until we feel that the problem won't recur !