According to a recent article that appeared in the June 1998 Business Traveller, this is one of the simpler ways to secure an upgrade. The existing scheme provides confirmed upgrades and /or companion tickets for Amex Gold and Platinum cardholders booking full-fare tickets with selected airlines. It can save you thousands of pounds per trip and although it ends in August, it will be extended with modifications into 1999.
Deals include:
-Continental, Qantas and South African Airways which have business class companion fares (the Qantas deal covers flights from Frankfurt or Rome, but not London)
-Cathay Pacific, Emirates, Malaysian and United, which provide upgrades from business to first class
-Canadian, Continental and Emirates, which will upgrade from economy to business class

I hope that this helps!