I am in the R&D phase of developing an application, with the following key requirements:
I wanted to make a small edit to update this question, as google is directing to us to this question if you make a research on the subject of websql, localStorage and indexedDB. The edit was rejected, so I'm posting as a answer.
As others stated in their answers, IndexedDB lacked a bit before on support and on content on the web for it's documentation and specification.
But IndexedDB support has been vastly improved for mobile. It improved so much that the only browser who has no support or whatsoever, is Opera Mini, but it has only 0,34% of market usage.
As of 2015, I would suggest any developer to move to IndexedDB, as WebSQL has been deprecated and IE and Firefox stopped supporting it (that's more than 15% of market usage for these alone!) and SQLite is losing space rapidly to IndexedDB, which have now really good documentation sources, many. Some official as well!. Some IT companies are even motivating usage as well, like IBM.
I'm going to use it, and I have not met any problems so far. Safari has added support to it, and all major browsers as well. Go for it!
EDIT: A personnal addendum: I tried IndexedDB. I'm a Senior on my team, and IndexedDB syntax is just too messy and complex for small storage problems - I ended using localstorage to save some simple JSON data and parsing it when I need it. Way better for anyone on my team to get it (Me as well, of course!)