Clinton Devon Estates who own the Common currently only say on their web site, that the Common is a "public open space used by many hundreds of thousands of people each year for walking, cycling, horse riding and bird watching". Therefore they don't seem to have anything against anyone cycling on the Common.
Devon County Council's pages about Public Rights of Way.
UK mountain biker web site.
Cycle route web sites.
National cycling organisations.
Sustrans - a groups which supports cycle routes, etc. but doesn't yet have any over the Common.
To find the Common on other mapping systems, try the following latitude and longitude: 50.66116,-3.364148
Ordnance Survey information about cycling in Devon.