Is there a free cloud based database service I can use for an asp.net application (hosted online and not on my own server) and a winform desktop application (running on my pc).
I'd like to use one so that both apps can use the same database.
SQL Azure looks like a good solution but has no free option.
Have you tried Appharbor.com?
They have a number of free database providers including SQL Server, mySQL, Mongo and Cloudant. They'll also build and test your software providing it's hosted on Github, Bitbucket or Codeplex. If you don't care about that, they'll still give the connection string to work with your winforms apps.