I am a Chinese national holding a US greencard. I am planning on traveling to the UK and the Schengen area to visit France, Germany, and possibly some countries in between. I am applying for a UK visa and a single entry Schengen visa now. On my journey to the UK I am stopping in Copenhagen for a day. Do I need a transit visa for that? Should I apply for a multiple entry Schengen visa, or is that too much? My guess is a transit visa for Denmark and a separate single entry visa would be easier to apply for.
Karlson's anwer is correct in that if you stay 'airside' you do not need a visa. I would however like to point out that since Copenhagen's airport mostly services flights within Schengen, most of the airside area is going to be inaccessible to you. In fact, only a very small part of the airport will be accessible to you without clearing passport control. While I can imagine spending an entire day in Copenhagen's airport, I think I'd go stir-crazy if limited to that small area for more than an hour or two.
Consequently, I'd strongly advice you to obtain (if you can) a visa that allows you a little more leeway. This will then not only open up the rest of the airport, it will enable you to go into Copenhagen, which is just a 20 minutes train ride from the airport.
There is nothing that says you have to have a transit visa if you don't leave airside, however, if you do you will need another full fledged Schengen short term visa as Chinese nationals are required to. You can also review the list of all agreements regarding visa issuance by Denmark.
And if this is not enough you can visit the VFS Global site for China which doesn't provide for a separate transit visa like the one in India for example (though the application does).