I am in the R&D phase of developing an application, with the following key requirements:
IndexedDB is most likely the supported database of the future and it would be best to go with that instead of WebSQL. As Raymond pointed, it is best to refer to http://www.caniuse.com to see the current/future support in both desktop and mobile browsers.
Depending on the current needs of your solution, you might be fine with one of the many JavaScript libraries that are available which use the local storage and provide a query interface. One of the libraries, which has worked well for me is Lawnchair.