Does a typical mobile adapter draw current when no load is attached?
Yes, it will draw current. It will also consume energy (what you pay for on your electric bill).

Modern mobile chargers consume perhaps 0.01kWh/day according to this book.(see page 68 in the book, not PDF, numbering) - measurements below indicate that contemporary chargers tend to do 2-5x better that that.. 

That's not a lot- you can calculate what it costs and/or what percentage of your total energy usage it is. At $0.10/kWh, 0.005kWh/day represents about 20 cents a year. If it takes you 5 seconds per day to plug and unplug, you're receiving an after-tax ROI of 40 cents an hour for your time.  

The current draw may be rather higher than the energy consumption would imply because some of the current is reactive.. however it cannot be less. 

Here's an excerpt from a set of actual measurements, made by Ken Shirriff: 



As you can see, the ones made by responsible manufacturers are generally 200mW or less, which would imply a current draw of a mA or two (maybe 1/10 of that) depending on reactive power, and depending on whether it was connected to a 120VAC or 230VAC mains. 

As Professor MacKay concludes, I would say it's a waste of time to unplug a modern mobile charger for a person living normally in a developed country. There are far bigger energy fish to fry.