I don't have any experience with Azure and it's too expensive to move there some projects but I think it would be great just to host some databases there (9,99$/month for 1GB).
It's quite difficult to find information about this and I don't know if you can only use it from an "azure application" or if you have to pay the instance time anyway...
You can use SQL Azure both from Windows Azure compute instances (VM's) and external apps, such as those hosted in your data center or another hoster. You just need to add the IP address range to the SQL Azure server firewall.
You'll just have to pay for bandwidth going from the Windows Azure data center to your asp.net app (which is hosted elsewhere).
EDIT: Here's a link to pricing. In North America and Europe, bandwidth is $0.12 / GB outbound (free inbound). $0.19 / GB in Asia Pacific region.
Yes, as David Mentioned you could use SQL Azure for an Azure app as well as other external apps. So, client/application can connect to SQL Azure via ADO.Net or ODBC connection library. And so apart from .Net apps, SQL Azure can also be connected from Java or PHP app. You need to use the following version of the connection libraries:
Data provider for SQL Server from .Net framework 3.5 SP1 or above
Entity Framework from .Net framework 3.5 SP1 or above
Complete list can be found here: http://msdn.microsoft.com/en-us/library/windowsazure/ee336245.aspx#dlaps
For SQL Azure pricing info, go to: http://msdn.microsoft.com/en-us/library/windowsazure/ee621788.aspx
