Silverlight does not have offline database support neither does Flash. The offline database support is most likely a confusion between Adobe Flash Player and Adobe AIR.
Silverlight however does have what I call "uber cookies" (yes that's a Barnes technical term) where you have the ability to store packets of data in what we call lsolated Storage. Isolated Storage Provides safe access from the Silverlight client to the local machine's file system. Enables local storage and caching of data isolated to a particular user.
Q. Is it per domain or per page? And if someone visits my site in IE and I save something in Isolated Storage will they be able to retrieve that information if they visit using the FireFox browser? - John
A. It’s currently per application’s source domain. Yes you can share persist data between browsers as they all point to the one area (within the secure sandbox). This will allow folks to share “Form values” - for example - between two browsers, so in the event UserA leaves for whatever reason in BrowserA, they can resume the data in BrowserB.
There are likely to be some adjustments to IsoStore in Silverlight 2.0. Keep an eye on Silverlight.NET’s blogs for more information as it’s announced.